The Properties of L-Moments Compared to Conventional Moments

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The Properties of L-moments Compared to Conventional Moments

August 17, 2009 THE ISLAMIC UNIVERSITY OF GAZA DEANERY OF HIGHER STUDIES FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS

The Properties of L-moments Compared to Conventional Moments

PRESENTED BY Mohammed Soliman Hamdan

SUPERVISED BY Prof. Mohamed I Riﬃ.

A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF MATHEMATICS Dedication

To the spirit of my father...

To my mother

To my wife

To all knowledge seekers... Contents

Acknowledgments V

Abstract VI

Introduction 1

1 Preliminaries 5 1.1 Distribution Functions and Probability Density or Mass Functions . . . . . 6 1.2 Random Samples ...... 8 1.3 Estimators ...... 9 1.4 Moment and Moment Generating Functions ...... 10 1.5 Skewness and Kurtosis ...... 16 1.6 The Shifted Legendre Polynomials ...... 21 1.7 Order Statistics ...... 25

2 L-MOMENTS OF PROBABILITY DISTRIBUTIONS 28 2.1 Deﬁnitions and Basic Properties ...... 29 2.2 Probability Weighted Moments ...... 37 2.3 Relation of L-moments with Order Statistic ...... 40 2.4 Properties of L-moments ...... 46 2.5 L-skewness and L-kurtosis ...... 52 2.6 L-moments of a Polynomial Function of Random Variables ...... 54 2.7 Approximating a Quantile Function ...... 57

II 2.8 L-moments as Measures of Distributional Shape ...... 59 2.9 L-moments for some Distributions ...... 63 2.9.1 L-moments for Uniform Distribution ...... 64 2.9.2 L-moments for Exponential Distribution ...... 67 2.9.3 L-moments for Logistic Distribution ...... 70 2.9.4 L-moments for Generalized Pareto ...... 71

3 ESTIMATION OF L-MOMENTS 75 3.1 The rth Sample L-moments ...... 75 3.2 The Sample Probability Weighted Moments ...... 78 3.3 The rth Sample L-moment Ratios ...... 81 3.4 Parameter Estimation Using L-moments ...... 84

4 Estimation of the Generalized Lambda Distribution from Censored Data 88 4.1 The Family of Generalized Lambda Distribution ...... 89 4.2 PWMs and L-moments for GLD ...... 89 4.3 PWMs and L-moments for Type I and II Singly Censored Data ...... 95 4.3.1 Case I-Right Censoring ...... 95 4.3.2 Case 2 - Left Censoring ...... 102 4.4 L-moments for Censored Distributions Using GLD ...... 105 4.5 Fitting of the Distributions to Censored Data Using GLD ...... 118

List of Tables Table 1.1 The Skewness For Some of Common Distributions ...... 18 Table 1.2 The Kurtosis For Some of Common Distributions ...... 20 Table 2.1 L-skewness of Some Common Distributions ...... 53 Table 2.2 L-kurtosis of Some Common Distributions ...... 53 R 1 −1 m k Table 2.3 Matrix B with Numerical Evaluations of βk = 0 (Φ (u)) u du . . . 55 R 1 m k Table 2.4 Matrix B with Numerical Evaluation of βk = 0 (ξ −α log(1−u)) u du 56 Table 2.5 L-moments of Some Common Distributions ...... 64

III Table 3.1 Annual Maximum Windspeed Data, in Miles per Hour ...... 82 Table 3.2 L-moments of The Annual Maximum Windspeed Data in Table(3.1) 83 Table 3.3 Bais of Sample L-CV ...... 83 Table 3.4 Parameter Estimation via L-moments for Some Common Distributions 85 Table 4.1 Comparison of L-moments ...... 94 Table 4.2 L-moment of Pareto Distribution for Censoring Fraction c ...... 117

IV Acknowledgments

I would like to express my sincere thanks and gratitude to Almighty Allah for his blessings. I am extremely and sincerely thankful to my parents whose love, care and sacrifice enabled me to reach this level of learning. I would like to express my sincere appreciation and thanks to my supervi- sor Prof. Mohamed I Riffi for his ceaseless help and supervision during the preparation of this project. I would like also to express my great and sincere thanks to Prof. Eissa D. Habil for his great help and sincere guidance and advice all the time. At the same time, I would like to thank Dr. Raid Salha for his great efforts with me. I would like to express my sincere thanks to all the staff members of the Mathematics Department and all my teachers who taught me to come to this stage of learning.

V Abstract

In this thesis, we survey the concept of L-moments. We introduce the definition of L-moments and the probability weighted moments (PWMs) and then we expressed the L-moments by the use of the probability weighted moments. Also, we established the relation between the L-moments and the order statistic. Moreover, we introduced some of the properties the L-moments especially the property that, if the mean of the distribution exists, then all of the L-moments exist and uniquely define the distribution. That is, no two distinct distributions have the same L-moments. This property is not always valid in the conventional moments. Moreover, we find the L-moments for some distributions. Later, we introduce estimation for the L-moments and probability weighted moments and then we used them in estimating the parameters of some distributions as the Uniform distribution, the Exponential distribution, Generalized Logistic distribution and Generalized Pareto Distribution. Moreover, we introduce the generalized lambda distribution (GLD) and we find the (PWMs) and L-moments for (GLD). Also, we defined the Censored Data which is divided into two cases: I-Right censoring and II-Left censoring and then we find the partial property weighted moments (PPWMs) for both cases. Finally, we find the type B PPWMs for GLD. Key words: Order Statistics, Probability Weighted Moments, L-moments, Censored Data, Generalized Lambda Distribution Family, Partial Probability Weighted Moments.

VI Introduction

It is standard statistical practice to summarize a probability distribution or an observed data set by its moments or cumulant. It is also common, when fitting a parametric distribution to a data set, to estimate the parameters by equating the sample moments to those of the fitted distribution. Yet moment-based methods, although long established in statistics, are not always satisfactory. It is sometimes difficult to assess exactly what information about the shape of a distribution is conveyed by its moments of third and higher order; the numerical values of sample moments, particularly when the sample is small, can be very different from those of the probability distribution from which the sample was drawn; and the estimated parameters of distributions fitted by the method of moments are often markedly less accurate than those obtainable by other estimation procedures such as the method of maximum likelihood. The alternative approach described here is based on quantities which we call L-moments. These are analogous to the conventional moments but can be estimated by linear combinations of order statistics. L-moments have theoretical advantages over conventional moments of being able to characterize a wider range of distributions and, when estimated from a sample, of being more robust to the presence of outliers in the data. Experience also shows that, compared with conventional moments, L-moments are less subject to bias in estimation and approximate their asymptotic normal distribution more closely in finite samples. Parameter estimates obtained from L-moments are sometimes more accurate in small samples than even the maximum likelihood estimates[17]. The origins of our work can be traced to the early 1970, when there was a growing awareness among hydrologists that annual maximum streamflow data, although commonly

1 modeled by the Gumbel distribution, often had higher skewness than was consistent with that distribution. Moment statistics were widely used as the basis for identifying and fitting frequency distributions, but to use them effectively required knowledge of their sampling properties in small samples. A massive (for the time) computational effort using simulated data was performed by Wallis, Matalas, and Slack in 1974. It revealed some unpleasant properties of moment statistics-high bias and algebraic boundedness. Wallis and others went on to establish the phenomenon of “separation of skewness,” which is that for annual maximum streamflow data “the relationship between the mean and the standard deviation of regional estimates of skewness for historical flood sequences is not compatible with the relations derived from several well-known distribution” (Matalas, Slack, and Wallis in 1975). Separation can be explained by “mixed distribution” (Wal- lis, Matalas, and Slack in 1977)- regional heterogeneity in our present terminology- or if the frequency distribution of stremflow has a longer tail than those of the distribution commonly used in the 1970s. In particular, the Wakeby distribution dose not exhibit the phenomenon of separation (Landwehr, Matalas, and Wallis in 1978). The Wakaby distribution was devised by H.A Thomas Jr. (personal communication to J.R. Wallis, in 1976). It is hard to estimate by conventional methods such as maximum likelihood or the method of moments, and the desirability of obtaining closed-from estimates of Wakeby parameters led Greenwood et al. (1979) to devise probability weighted moments. Prob- ability weighted moments were found to perform well for other distributions (Landwehr, Matalas, and Wallis in 1979; Hosking, Wallis, and Wood in 1985; Hosking and Wallis in 1987) but were hard to interpret. In 1990 Hosking found that certain linear combinations of probability weighted moments, which he called “L-moments,” could be interpreted as measures of the location, scale, and shape of probability distribution and formed the basis for a comprehensive theory of the description, identification, and estimation of distributions ([15], Pages xi, xii). This thesis consists of four chapters. In the first chapter, we introduce general concepts and definitions that are related to the L-moments. The definition of the cumulative distribution function, quantile function and the probability density function are very important

2 in chapter two. The definition of the random sample is essential in the definition of the order statistic. The concept of the estimator is useful in chapter three in the estimation of L-moment. The concepts of the nth moment, rth central moment and moment generated functions are introduced to be used in comparing between the conventional moments and L-moments. The concept of order statistics is the base for defining the L-moments. In fact, the first chapter consists of seven sections: distribution functions and probability density or mass functions, random samples, estimators, moment and moment generating functions, skewness and kurtosis, the shifted Legendre polynomials and order statistics. Charter 2, which is the main chapter in this research, consists of nine sections. In this chapter, we define L-moments and L-moments ratios in the first section. In the second section, we define probability weight moments and we find the relationship between L- moments and probability weight moments and it will make it easier to find L-moments for some distributions. In the third section, we find the relation between L-moments and order statistic. In the fourth section, we establish some properties of L-moments. After that, we talk about L-skewness and L-kurtosis (which are considered as a special case of the L-moments ) in section 2.5. In the sixth section, we write about the L-moments of a polynomial function of a random variable. In the seventh section, we write about an inversion theorem, expressing the quantile function in terms of L-moments. In the eighth section, we write about L-moments as a measure of distribution shape. Finally, in the ninth section, we find L-moments for some distributions. This section is divided into four subsections: L-moments for uniform distribution, L-moments for exponential distribution, L-moments for logistic distribution and L-moments for generalized pareto distribution. This last section is used in chapter three in estimating the parameters of some of the previous distributions. Chapter 3, which is titled by estimation of L-moments, consists of four sections: the rth sample L-moments (which is used in estimating the parameters of some distributions), the sample probability weighted moments (which is used in chapter four in finding PP- WMs estimators for Right and Left Censoring), the rth sample L-moment ratios, and finally the parameter estimation using L-moments.

3 In chapter 4, we deal with the “Estimation of the Generalized Lambda Distribution from Censored Data”. In the first section, we find the PWMs and L-moments for GLD. In the second section, we discus the PWMs and L-moments for Censored Data (type B for Right Censoring and Left Censoring ). In the third section, we find L-moments for Censored Distributions using GLD. In the last section, we discuss the fitting of the distributions to Censored Data using GLD. In fact, chapter 4 is considered as an application of the previous chapters.

4 Chapter 1

Preliminaries

In this chapter, we give the basic definitions, that we think they are very important for our thesis. In the first section, we define the cumulative distribution functions, quantile functions and the probability density functions, and these definitions are needed in chapters 2, 3 and 4. In the second section, we define the random sample and give related examples. The importance of section two will appear in section (1.7). In the third section, we wrote about estimators and define bias estimators. This section is necessary in chapter three. In section four, we define the nth moment and the nth central moment and find the nth center moment for normal distribution. After that, we define skewness and kurtosis in section five. In the sixth section we define the shifted Legendre polynomials. Finally, we introduce the order statistic and its distributions in section seven.

5 1.1 Distribution Functions and Probability Density or Mass Functions

In this section we define the cumulative distribution functions, quantile functions and the probability density functions. These definitions are essential in defining the L-moments, the main definition in this research.

Deﬁnition 1.1.1. ([26], Page 112) Let X be random variable deﬁned on a sample space S with probability function P . For any real number x, the cumulative distribution function of X [abbreviated ( cdf ) and written F (x)] is the probability associated with the set of sample points in S that get mapped by X into values on the real line less than or equal to x. Formally, F (x) := P ({s ∈ S | X(s) ≤ x}).

We shall normally be concerned with continuous random variables, F (x) is an increasing function of x, and 0 ≤ F (x) ≤ 1 for all x, for which P (X = t) = 0 for all t. That is; no single value has nonzero probability. In this case, F (x) is a continuous function and has an inverse function.

Deﬁnition 1.1.2. ([15], Page 14) If F (x) is the cumulative distribution function of X, then the inverse function of F (x) is called the quantile function of X and is denoted by x(F ).

Notice that, given any u, 0 < u < 1, x(u) is the unique value that satisﬁes

F (x(u)) = u.

Deﬁnition 1.1.3. ([10], Page 35) The probability density function (pdf) of a continuous random variable X is the function f satisfying

Z x F (x) := f(t)dt for all x. −∞ Remark 1.1.1. We deduce from the above two deﬁnitions the following: P P 1. If X is a discrete random variable, then F (x) = y≤x P (X = y) = y≤x f(y), and in this case, f(x) is said to be probability mass function (pmf) of X.

6 2. If X is a continuous random variable, and f is a continuous function, then by the

d Fundamental Theorem of Calculus, f(x) = dx F (x).

Deﬁnition 1.1.4. ([26], Page 131) Two random variables X and Y are said to be independent, if and only if fXY (x, y) = fX (x)fY (y), for all x and y where f(x, y) is the joint

(pdf) or (pmf) of X and Y , and f(x)X , f(y)Y are the (pdf) of X and Y , respectively.

Deﬁnition 1.1.5. ([10], Page 174) Let X1,X2, ..., Xn be random variables with the joint

(pdf) or (pmf) f(x1, x2, ..., xn). Let fi(x) denote the marginal (pdf) or (pmf) of Xi Then

X1,X2, ..., Xn are called mutually independent random variables if for every (x1, x2, ..., xn) within their range Yn f(x1, x2, ..., xn) = fi(xi). i=1 Deﬁnition 1.1.6. ([26], Page 154) Let X be any random variable with the marginal (pdf) or (pmf) f(x). The expected value denoted by E(X) and is given by:

Z ∞ Z ∞ (1) E(X) = xf(x) dx; if X is a continuous random variable, provided that |x|f(x) < ∞ . −∞ −∞ (1.1.1) We may also write, via the transformation u = F (x),

Z 1 E(X) = x(u)du . 0 P P (2) E(X) = x xf(x) if X is a discrete random variable, provided that x |x|f(x) < ∞ .

Example 1.1.1. Let X be a random variable from the exponential distribution with parameter β. Then the expectation of X is given by:

Z ∞ E(X) = xf(x)dx −∞ Z ∞ ³ ´ 1 −x = x e β dx 0 β = β.

7 1.2 Random Samples

In this section, we deﬁne the random sample which is used to deﬁne the order statistics in section 1.7. Then, we give related examples.

Deﬁnition 1.2.1. ([10], Page 201) The collection of random variables X1,X2, ..., Xn is called a random sample of size n from the population with (pdf) f(x) if X1,X2, ..., Xn are mutually independent and marginal probability density function (pdf) or probability mass function (pmf) of each Xi is the sample function f(x).

Alternatively, X1,X2, ..., Xn are called independent and identically distributed random variables with (pdf) or (pmf) f(x). This is commonly abbreviated to iid random variables. From the above deﬁnition of a random sample, the joint (pdf) or (pmf) of the random sample X1,X2, ..., Xn is given by Yn f(x1, x2, ..., xn) = f(x1)f(x2)...f(xn) = f(xi). i=1

Example 1.2.1. Let X1,X2, ..., Xn be a random sample of size n from the exponential distribution with parameter (β), corresponding to the time until failure for identical circuit that one puts on the test and used until they fail. Then the joint (pdf) of the sample is:

f(x1, x2, ..., xn|β) = f(x1)f(x2)...f(xn) Yn = f(xi|β) i=1 Yn − xi = (1/β)e β i=1 P n − 1 n x = (1/β) e β i=1 i .

8 Now, to compute the probability of the all boards last more than 2 time units, we do the following Yn P (X1 > 2,X2 > 2, ..., Xn > 2) = P (Xi > 2) i=1 Z Yn ∞ 1 −xi/β = e dxi 2 β i=1Z ³ ∞ 1 ´n = e−x/βdx 2 β = (e−2/β)n = e−2n/β .

1.3 Estimators

In practice, it is often assumed that the distribution of some physical quantities is exactly known apart from a ﬁnite set of parameters θ1, ...., θp. When needed for clarity, we write the quantile function of a distribution with p unknown parameters as x(u; θ1, ...., θp). In most applications the unknown parameters include a location parameter and a scale parameter [15].

Deﬁnition 1.3.1. ([15], Page 15) A parameter ξ of a distribution is a location parameter if the quantile function of the distribution satisﬁes

x(u; ξ, θ1, ...., θp) = ξ + x(u; 0, θ1, ...., θp).

Deﬁnition 1.3.2. ([15], Page 16) A parameter α of a distribution is a scale parameter if the quantile function of the distribution satisﬁes

x(u; α, θ1, ...., θp) = α × x(u; 1, θ1, ...., θp). or, if the distribution also has a location parameter ξ,

x(u; ξ, α, θ1, ...., θp) = ξ + α × x(u; 0, 1, θ1, ...., θp).

Example 1.3.1. The gamble distribution has the quantile function[15]:

x(u) = ξ − α log(− log u).

9 Since x(u; ξ, α) = (ξ) + [−α log(− log u)] = ξ + x(u; 0, α), then ξ is a location parameter. Now, ξ is a location parameter and x(u; ξ, α) = ξ−α log(− log u) = (ξ)+(α)[− log(− log u)] = ξ + α × x(u; 0, 1), hence α is a scale parameter.

The unknown parameters are estimated from the observed data. Given a set of data, a function θˆ of the data values may be chosen as an estimator of θ. The estimator θˆ is a random variable and has a probability distribution. The goodness of θˆ as an estimator of θ depends on how close θˆ typically is to θ. The deviation of θˆ from θ may be decomposed into bias - a tendency to give estimates that are consistently higher or lower than the true value - and variability - the random deviation of the estimate from the true value that occurs even for estimators that have no bias [15].

Deﬁnition 1.3.3. ([15], Page 16) bias(θˆ) = E(θˆ − θ)

Deﬁnition 1.3.4. ([15], Page 16) We say that θˆ is unbiased if bias(θˆ) = 0, that is if E(θˆ) = θ.

1.4 Moment and Moment Generating Functions

In this section, we define the nth moment, nth central moment and also define the moment generating function. Also, we introduce a theorem that generates the moment from moment generating function and find the nth center moment for normal distribution. After that, we define skewness and kurtosis. The shape of a probability distribution has traditionally been described by the moments of the distribution.

0 Deﬁnition 1.4.1. ([10], Page 58) For each integer n, the nth moment of X, µn, is

0 n µn = E(X ).

The nth central moment of X, µn, is

n µn = E(X − µ) ,

0 where µ = µ1 = E(X).

10 The mean is the center of location of the distribution. The dispersion of the distribution about its center is measured by the standard deviation,

1/2 2 1/2 σ = µ2 = {E(X − µ) } ,

2 or the variance, σ = var(X). The coeﬃcient of variation (CV), Cv = σ/µ,

Deﬁnition 1.4.2. ([15], Page 17) Analogous quantities can be computed from a data sample x1, x2, ..., xn. The sample mean Xn −1 x¯ = n xi i=1 is the natural estimator of µ.

Deﬁnition 1.4.3. ([15], Page 17) The higher sample moments Xn −1 r mr = n (xi − x¯) i=1 are reasonable estimators of the µr, but are not unbiased.

2 Unbiased estimators are often used. In particular, σ , µ3 and the fourth cumulant 2 κ4 = µ4 − 3µ2 are unbiasedly estimated by Xn 2 −1 2 s = (n − 1) (xi − x¯) , i=1 n2 m˜ = m 3 (n − 1)(n − 2) 3 n2 n³n + 1´ o k˜ = m − 3m2 , 4 (n − 2)(n − 3) n − 1 4 2 √ respectively. The sample standard deviation, s = s2, is an estimator of σ but is not unbiased. The sample estimator of CV, is,

ˆ Cv = s/x¯

We now introduce a new function that is associated with a probability distribution, the moment generating function mgf. As its name suggests, the mgf can be used to generate moments.

11 Deﬁnition 1.4.4. ([15], Page 61) Let X be a random variable with cdf F (X). The moment generating function (mgf ) of X, denoted by MX (t), is

tX MX (t) = E(e ) , provided that the expectation exists for t in some neighborhood of 0.

More explicitly, we can write the mgf of X as

Z ∞ tx MX (t) = e f(x)dx if X is continuous −∞ or X tx MX (t) = e P (X = x) if X is discrete. x It is very easy to see how the mgf generates moments. We summarize the result in the following theorem.

n (n) Theorem 1.4.1. [15] If X has mgf MX (t), then E(X ) = MX (0), where we deﬁne

n ¯ (n) d ¯ MX (0) = MX (t)¯ . dtn t=0 th th That is, the n moment is equal to the n derivative of MX (t) evaluated at t = 0.

Proof. Assuming that X has (pdf) fX (x). If we can diﬀerential under the integral sign we have

Z ∞ d d d tx MX (t) = MX (t) = e fX (x)dx dt dt dt −∞ Z ∞ d tx = ( e )fX (x)dx −∞ dt Z ∞ tx = (xe )fX (x)dx −∞ = E(XetX ).

12 Thus, d M (t)| = E(XetX ) | = E(X). dt X t=0 t=0 Proceeding in an analogous manner, we can establish that

dn M (t)| = E(XnetX ) | = E(Xn). dtn X t=0 t=0

Deﬁnition 1.4.5. ([10], Page 100) For any real number r > 0, the gamma function (of r) is given by: Z ∞ Γ(r) = xr−1e−xdx. 0 Note 1.4.2. ([10], Page 100) If r is a positive real number, then Γ(r + 1) = rΓ(r).

Note 1.4.3. ([10], Page 100) For any positive integer n, Γ(n) = (n − 1)! ·

Example 1.4.1. The full gamma(α, β) family, is,

1 f(x) = xα−1e−x/β, 0 < x < ∞, α > 0, β > 0, Γ(α)βα where Γ(α) denotes the gamma function,

Z ∞ 1 tx α−1 −x/β MX (t) = α e x e dx Γ(α)β 0

Z ∞ 1 1 α−1 −( β −t)x = α x e dx (1.4.1) Γ(α)β 0

Z ∞ 1 β α−1 −x/( 1−βt ) = α x e dx. Γ(α)β 0 Using the fact that, for any positive constants a and b,

1 f(x) = xa−1e−x/b Γ(a)ba

13 is a pdf, we have that Z ∞ 1 a−1 −x/b a x e dx = 1 0 Γ(a)b and hence, Z ∞ xa−1e−x/bdx = Γ(a)ba. (1.4.2) 0 Applying (1.4.2) to (1.4.1), we have

1 ³ β ´α ³ 1 ´α 1 M (t) = Γ(α) = if t < . X Γ(α)βα 1 − βt 1 − βt β 1 If t ≥ β , then the quantity (1/β) − t, in the integrand of (1.4.1), is nonpositive and the integral in (1.4.2) is inﬁnite. Thus, the mgf of the gamma distribution exists only if t < 1/β The mean of the gamma distribution is given by

¯ d ¯ αβ ¯ EX = MX (t)¯ = ¯ = αβ. dt t=0 (1 − βt)α+1 t=0

Other moments can be calculated in similar manner.

Example 1.4.2. Central moments of the normal distribution N(0,σ2). The moment generating function for the normal distribution N(0,σ2) is as follows:

t2σ2 MX (t) = e 2 .

The moments are then as follows. The ﬁrst central moments is

³ 2 2 ´ d t σ 2 E(X − µ) = dt e |t=0

³ 2 2 ´ 2 t σ = tσ e 2 |t=0

= 0.

14 The second central moment is ³ 2 2 ´ d2 t σ 2 2 E(X − µ) = dt2 e |t=0

³ ³ 2 2 ´´ d t σ 2 2 = dt tσ e |t=0

³ ³ 2 2 ´ ³ 2 2 ´´ 2 4 t σ 2 t σ = t σ e 2 + σ e 2 |t=0

= σ2.

The third central moment is ³ 2 2 ´ d3 t σ 3 2 E(X − µ) = dt3 e |t=0

³ ³ 2 2 ´ ³ 2 2 ´´ d t σ t σ 2 4 2 2 2 = dt t σ e + σ e |t=0

³ ³ 2 2 ´ ³ 2 2 ´ ³ 2 2 ´´ 3 6 t σ 4 t σ 4 t σ = t σ e 2 + 2tσ e 2 + tσ e 2 |t=0

³ ³ 2 2 ´ ³ 2 2 ´´ 3 6 t σ 4 t σ = t σ e 2 + 3tσ e 2 |t=0

= 0.

The fourth central moment is ³ 2 2 ´ d4 t σ 4 2 E(X − µ) = dt4 e |t=0

³ ³ 2 2 ´ ³ 2 2 ´´ d t σ t σ 3 6 2 4 2 = dt t σ e + 3tσ e |t=0

³ ³ 2 2 ´ ³ 2 2 ´ ³ 2 2 ´ ³ 2 2 ´´ 4 8 t σ 2 6 t σ 2 6 t σ 4 t σ = t σ e 2 + 3t σ e 2 + 3t σ e 2 + 3σ e 2 |t=0

³ ³ 2 2 ´ ³ 2 2 ´ ³ 2 2 ´´ 4 8 t σ 2 6 t σ 4 t σ 4 = t σ e 2 + 6t σ e 2 + 3σ e 2 |t=0 = 3σ .

15 Now, we write this Theorem because it is used in the proof of Theorem 2.4.1.

Theorem 1.4.4. ([10], Page 65) Let FX (x) and FY (y) be two cdfs all whose moments exist. If FX (x) and FY (y) have bounded support, then FX = FY for all u if and only if EXr = EY r for all integers r = 0, 1, 2,...

Proof. Assume that FX (u) = FY (u) for all u, hence dFX (u) = dFY (u). Now, for all integers r = 0, 1, 2,...,

Z ∞ Z ∞ r r r r E(X ) = u dFX (u) = u dFY (u) = E(Y ). −∞ −∞ Conversely, assume that EXr = EY r for all integers r = 0, 1, 2,... , then in special case EX0 = EY 0. Conceder, Z ∞ Z ∞ 0 0 EX = u dFX (u) = dFX = FX . −∞ −∞ Similarly, Z ∞ Z ∞ 0 0 EY = u dFY (u) = dFY = FY . −∞ −∞ 0 0 Since EX = EY , then FX = FY . That means, FX (u) = FY (u), for all u.

1.5 Skewness and Kurtosis

Skewness measures the lak of symmetry in the probability density function f(x) of a distribution [10].

Deﬁnition 1.5.1. ([15], Page 17) The skewness is :

3/2 γ = µ3/µ2 .

A distribution that’s symmetric about its mean has 0 skewness. But if it has a long tail to the right and a short one to the left, then it has a positive skewness, and a negative skewness in the opposite situation. The sample estimator of skewness is,

3 g =m ˜ 3/s [15],

16 where Xn 2 −1 2 s = (n − 1) (xi − x¯) , i=1 n2 m˜ = m . 3 (n − 1)(n − 2) 3

The estimator g is biased estimators of γ. Indeed, g has algebraic bounds that depend on the sample size; for a sample of size n the bound is

|g| ≤ n1/2 [15].

Example 1.5.1. The skewness of the normal distribution N(0,σ2): From example 1.4.2, we have the second and the third central moments of the normal dis-

2 2 tribution N(0,σ ) are: µ2 = σ and µ3 = 0. Then, the skewness of the normal distribution N(0,σ2) is: 0 0 γ = µ /µ3/2 = = = 0 . 3 2 (σ2)3/2 σ3

17 Table 1.1: The following table gives the skewness for a number of common distributions. Distribution pdf, f(x) Skewness

Bernoulli pxq1−x √1−2p p(1q−p) Γ(α+β) β−1 α 2(β−α) 1+α+β Beta Γ(α)Γ(β) (1 − x) x (2+α+β) αβ ¡ ¢ Binomial N pxqN−x √q−p x qNpq xr/2−1 e−x/2 2 Chi-squared 1 r/2 2 r Γ( 2 r) 2 1 −(x−α)/β Exponential β e 2 α−1 −x/θ x e √2 Gamma Γ(α) θα α

x √2−p Geometric distribution p q 1−p √ 2θ −x2θ2/π 2(4−n) Half-normal π e (π−2)3/2 1 −|x−µ|/b Laplace 2b e 0 2 2 √ 2 Log normal √1 e−(ln x−M) /(2S ) eS2 − 1(2 + eS ) S 2π x q 2 2 √ 2 x2e−x /(2a ) 2 2(5n−16) Maxwell π a3 (3n−8)3/2 ¡x+r−1¢ 2−p r x √ Negative binomial r−1 p q rq 2 2 Normal √1 e−(x−µ) /(2σ ) 0 σ 2π νn e−ν −1/2 Poisson n! ν 2 2 q x e−x /(2s ) π Rayleigh s2 (π − 3) 1 3 2(2− 2 π) ¡ ¢(1+r)/2 r r+x2 Student’s t √ 1 1 0 r B( 2 r, 2 ) 1 Continuous uniform β−α 0 1 Discrete uniform N 0

18 Kurtosis kurtosis is the degree of peakedness of a distribution, deﬁned as a normalized from the fourth central moment µ4.

Deﬁnition 1.5.2. ([15], Page 17) The kurtosis is

2 κ = µ4/µ2.

A fairly ﬂat distribution with long tails has a high kurtosis, while a short tailed distribution has a low kurtosis. A normal distribution has a kurtosis of 3. The sample estimators of kurtosis,

˜ 4 k = k4/s + 3 [15], where Xn 2 −1 2 s = (n − 1) (xi − x¯) , i=1 n2 n³n + 1´ o k˜ = m − 3m2 . 4 (n − 2)(n − 3) n − 1 4 2

The estimator k is biased estimators of κ. Indeed k has algebraic bounds that depend on the sample size; for a sample of size n the bound is

k ≤ n + 3 [15].

Example 1.5.2. The kurtosis of the normal distribution N(0,σ2): Since the second and the fourth central moments of the normal distribution N(0,σ2) are:

2 4 µ2 = σ and µ4 = 3σ (see example 1.4.2). Hence, the kurtosis of the normal distribution N(0,σ2) is: 3σ4 3σ4 κ = µ /µ2 = = = 3 . 4 2 (σ2)2 σ4

19 Table 1.2: The following table gives the Kurtosis for a number of common distributions. Distribution pdf, f(x) Kurtosis

x 1−x 1 1 Bernoulli p q 1−p + p − 6 Γ(α+β) β−1 α 6[a3+a2(1−2b)+b2(1+b)−2ab(2+b)] Beta Γ(α)Γ(β) (1 − x) x ab(2+a+b)(3+a+b) ¡ ¢ N x N−x 1−6pq Binomial x p q Npq xr/2−1 e−x/2 12 Chi-squared 1 r/2 r Γ( 2 r) 2 1 −(x−α)/β Exponential β e 6 xα−1e−x/θ 6 Gamma Γ(α) θα α x 1 Geometric distribution p q 5 − p + 1+p 2θ −x2θ2/π 8(π−3) Half-normal π e (π−2)2 1 −|x−µ|/b Laplace 2b e 3 2 2 2 3S2 2S2 Log normal √1 e−(ln x−M) /(2S ) e4S +2e +3e −6 S 2π x q 2 2 2 x2e−x /(2a ) 4(96−40π+3π2) Maxwell π a3 − (3π−8)2 ¡ ¢ x+r−1 r x 6−p(6−p) Negative binomial r−1 p q r(1−p) 2 2 Normal √1 e−(x−µ) /(2σ ) 3 σ 2π νn e−ν 1 Poisson n! v 2 2 x e−x /(2s ) 6π(4−π)−16 Rayleigh s2 (π−4)2 ¡ ¢(1+r)/2 r 2 √r+x 6 Student’s t 1 1 r−4 r B( 2 r, 2 ) 1 6 Continuous uniform β−α − 5 1 6(N 2+1) Discrete uniform N − 5(N 2−1)

20 1.6 The Shifted Legendre Polynomials

th The base of our thesis is to deﬁne the L-moments λr which depends on the r shifted

Legendre polynomial which is related to the usual Legendre polynomials Pr−1(F ). So, we deﬁned the Legendre polynomials and the shifted Legendre polynomials and we extract some relations that we use in this thesis. In addition, we show that Legendre polynomials and shifted Legendre polynomials are eigenfunctions. Furthermore, we serve the Corollary that will be used to prove Theorem 2.7.1 in section (2.7).

Deﬁnition 1.6.1. ([5], Page 60) A self-adjoint diﬀerential equation of the form

[p(x) y0]0 + [q(x) + λr(x)] y = 0, (1.6.1) on the interval 0 < x < 1, together with the boundary conditions

0 0 a1y(0) + a2y (0) = 0, b1y(1) + b2y (1) = 0, (1.6.2) is called a Sturm-Liouville eigenvalue problem. Those values of λ for which non-trivial solutions for such problems exits, are called eigenvalues and the corresponding solutions are called eigenfunctions.

The following theorem expresses the property of orthogonality of the eigenfunctions with respect to the weight function r.

Theorem 1.6.1. ([31], Page 636) If y1 and y2 are two eigenfunctions of a Sturm-Liouville problem (1.6.1), (1.6.2) corresponding to eigenvalues λ1and λ2, respectively, and λ1 6= λ2, then Z 1 r(x) y1(x) y2(x) dx = 0, where r(x) is weight function. (1.6.3) 0

21 Corollary 1.6.2. ([5], Page 61) (Eigenfunction expansion). If {yi(x)} is the set of eigenfunctions of the Sturm-Liouville eigenvalue problem:

[p(x) y0]0 + [q(x) + λr(x)] y = 0,

0 0 a1y(a) + a2y (a) = 0, b1y(b) + b2y (b) = 0, and f(x) is a function on [a,b] such that f(a) = f(b) = 0, then X∞ f(x) = ci yi(x) (1.6.4) i=0

Z b Z b 1 2 where ci = r(x) f(x) yi(x) dx, µi = r(x) yi (x) dx. µi a a Deﬁnition 1.6.2. ([5], Page 83) The Legendre’s equation is

(1 − x2) y00 − 2xy0 + n(n + 1) y = 0 (1.6.5) where n is a positive integer.

One of the solutions of equation (1.6.5) is the polynomial

h 1 − xi P (x) = F − n, n + 1; 1; , n 2 where Z h i Γ(c) 1 F a, b; c; x = tb−1(1 − t)c−b−1(1 − xt)−a dt Γ(b)Γ(c − b) 0 and Γ(.) is gamma function.

th Pn(x) is called the n Legendre’s polynomial.

The Legendre’s equation can be written in the self-adjoint form

[(1 − x2) y0]0 + n(n + 1) y = 0 (1.6.6)

Comparing equation (1.6.6) with the form (1.6.1), p(x) = 1−x2, q(x) = 0, r(x) = 1, λ = n(n + 1). Since p(x) = 0 for x = −1, 1, represents a Sturm-Liouville problem without explicit boundary conditions, its eigenfunctions are Pn(x) with the related eigenvalues

22 n(n + 1) (n = 0, 1, 2,...). Hence {Pn(x)} is an orthogonal set of polynomials over the interval −1 6 x 6 1 with weight function equal to 1, i.e.,

Z 1 Pm(x)Pn(x) dx = 0, m 6= n. −1 There are other approaches for establishing orthogonality of the Legendre sequence. The following is the complete statement [5]  Z 1  0, m 6= n; Pm(x)Pn(x) dx = −1  2 2n+1 , m = n.

0 Definition 1.6.3. ([15], Page 19) We define polynomials Pr(u), r = 0, 1, 2,... as follows: 0 (i) Pr(u) is a polynomial of degree r in u. 0 (ii) Pr(1) = 1. R 1 0 0 (iii) 0 Pr(u)Ps(u) du = 0 if r 6= s. Condition(iii) is the orthogonality condition. These conditions define the shifted Legendre polynomials Condition(“ shifted”, because the ordinary Legendre polynomials Pr(u) are defined to be orthogonal on the interval −1 ≤ u ≤ +1, not 0 ≤ u ≤ 1).

0 th The Pr(F ) is the r shifted Legendre polynomial related to the usual Legendre poly- 0 nomials Pr(u) = Pr(2u − 1). Shifted Legendre polynomials are orthogonal on the interval (0,1) with constant weight function r(u) = 1[15].

∗ th Note 1.6.3. [18] The Pr (F ) is the r shifted Legendre polynomials, where Xr ∗ ∗ m Pr (F ) = pr,m F , m=0 and     r r + m ∗ r−m     pr,m = (−1) . m m R 1 ∗ 2 1 Note 1.6.4. 0 {Pr (u)} du = 2r+1 ·

Proof. Since, Z 1 Z 1 ∗ 2 2 {Pr (u)} du = {Pr(2u − 1)} du. (1.6.7) 0 0

23 Let z = 2u − 1, then, dz = 2du and substituting in eqn (1.6.7) we have:

Z 1 Z 1 ³ ´ ∗ 2 1 2 1 2 1 {Pr (u)} du = {Pr(z)} dz = = · (1.6.8) 0 2 −1 2 2r + 1 2r + 1

R 1 ∗ Note 1.6.5. 0 Pr (u)du = 0 for r > 0.

∗ Proof. From deﬁnition of Pr (u), we have     X0 0 0 ∗ ∗ m ∗ 0−0     P0 (u) = p0,m u = p0,0 = (−1) = 1. m=0 0 0

Then from orthogonality condition,

Z 1 Z 1 Z 1 ∗ ∗ ∗ ∗ Pr (u)du = 1 × Pr (u)du = P0 (u) Pr (u)du = 0 becase r > 0 · (1.6.9) 0 0 0

We introduce (Chebyshev’s Other Inequality) since it is used in the proof of Theorem 2.8.1.

Theorem 1.6.6. [9] (Chebyshev’s Other Inequality). Let f and g be real-valued functions that are either both increasing or both decreasing on the interval (a,b) (a and b can be inﬁnite), and let w be a function that is positive on (a,b). Then

Z b Z b Z b Z b f(x)g(x)w(x)dx w(x)dx ≥ f(x)w(x)dx g(x)w(x)dx. a a a a Proof. We have

{f(y) − f(x)}{g(y) − g(x)} ≥ 0 for any x and y in (a, b), so,

Z b Z b 0 ≤ {f(y) − f(x)}{g(y) − g(x)}w(x)w(y)dx dy a a Z b Z b Z b Z b = f(y)g(y)w(x)w(y)dx dy − f(y)g(x)w(x)w(y)dx dy a a a a

24 Z b Z b Z b Z b − f(x)g(y)w(x)w(y)dx dy + f(x)g(x)w(x)w(y)dx dy a a a a Z b Z b Z b Z b = f(y)g(y)w(y)dy w(x)dx − f(y)w(y)dy g(x)w(x)dx a a a a Z b Z b Z b Z b − f(x)w(x)dx g(y)w(y)dy + f(x)g(x)w(x)dx w(y)dy a a a a Z b Z b Z b Z b = 2 f(x)g(x)w(x)dx w(x)dx − 2 f(x)w(x)dx g(x)w(x)dx. a a a a The result follows.

1.7 Order Statistics

In this section, we deal with order statistics and related subjects. At first, we define order statistics and their distribution functions. Next, we give examples for order statistics . Then, we present some significant propositions. After that, we define the probability density function and the cumulative distribution function for an order statistic. We then present some related theorems.

Deﬁnition 1.7.1. ([10], Page 229) The order statistics of a random sample X1,X2, ..., Xn are the sample values placed in ascending order. They are denoted by X(1),X(2), ..., X(n).

In other words, the order statistics are random variables that satisfy X(1) ≤ X(2) ≤ ... ≤

X(n), where

X(1) := min Xi i nd X(2) := 2 smallest Xi .

X(n) := max Xi. 1≤i≤n

Example 1.7.1. The values x1 = 0.62, x2 = 0.98, x3 = 0.31, x4 = 0.81 and x5 = 0.53 are the n = 5 observed values of ﬁve independent trials of an experiment with ( pdf)

25 f(x) = 2x, 0 < x < 1. The observed values of the order statistics are

x1 = 0.31 < x2 = 0.53 < x3 = 0.62 < x4 = 0.81 < x5 = 0.98.

Now, the next theorem gives the cdf of the jth order statistic.

Theorem 1.7.1. ([10], Page 231) Let X1,X2, ..., Xn be a random sample of size n from a distribution with pdf f(x) and (cdf) F (x). Then the cdf of the jth order statistic, is given by   Xn n k n−k Fj(x) =   [F (x)] [1 − F (x)] . (1.7.1) k=j k

Example 1.7.2. Let X1,X2, ..., Xn be a random sample of size n from the uniform distribution with parameter θ. Then   1 , 0 < x < θ; f(x) = θ  0, otherwise.

and   0, x ≤ 0;  F (x) = x , 0 < x < θ;  θ   1, x ≥ θ.   Xn n k n−k Fj(x) =   [F (x)] [1 − F (x)] k=j k   n X n hxikh ³x´in−k =   1 − . θ θ k=j k

Example 1.7.3. Let X1,X2, ..., Xn be the random sample of size n from an exponential distribution with parameter β. Then  − x  1 e β , x ≥ 0; f(x) = β  0, otherwise.

So,

26   Xn n k n−k Fj(x) =   [F (x)] [1 − F (x)] k=j k   n X h ³ ´ikh in−k n − x − x =   1 − e β e β . k=j k

Now, we introduce the probability density function of any order statistic through the following theorem.

Theorem 1.7.2. ([10], Page 232) Let X1,X2, ..., Xn be a random sample of size n from a distribution of continuous population with (pdf)f(x) and cdf F (x). Then the (pdf) of the jth order statistic is given by  

n j−1 n−j fj(x) = j   f(x)[F (x)] [1 − F (x)] . (1.7.2) j

Example 1.7.4. Let X1,X2, ..., Xn be a random sample of size n from the uniform distribution with parameter θ = 1. Then by Example 1.7.2, the cdf is deﬁned by:   0, x ≤ 0,  F (x) = x, 0 < x < 1,    1, x ≥ 1.

Now, for 0 < x < 1 , Theorem 1.7.2 yields  

n j−1 n−j fj(x) = j   f(x)[F (x)] [1 − F (x)] j   n = j   xj−1(1 − x)n−j j Γ(n + 1) = xj−1(1 − x)(n−j+1)−1. Γ(j)Γ(n − j + 1) Thus, the jth order statistic has a Beta distribution with parameters j and n − j + 1 .

27 Chapter 2

L-MOMENTS OF PROBABILITY DISTRIBUTIONS

L-moments are expectations of certain linear combinations of order statistics. They can be defined for any random variable whose mean exists and from the basis of a general theory which covers the summarization and description of theoretical probability distributions, the summarization and description of observed data samples, estimation of parameters and quantile of probability distributions, and hypothesis tests for probability distributions [17]. In the first section of this chapter, we define L-moments and L-moment ratios. In the second section, we define probability weight moments and we find the relationship between L-moments and probability weight moments and it will make it easier to find L-moments for some distributions. In the third section, we find the relation between L-moments and order statistic. In the fourth section, we established some properties of L-moments. After that, we talk about L-skewness and L-kurtosis. In the sixth section, we write about the L-moments of a polynomial function of a random variable.

28 In the seventh section, we write about an inversion theorem, expressing the quantile function in terms of L-moments. In the eighth section, we write about L-moments as measure of distribution shape. Finally, in the ninth section, we ﬁnd L-moments for some distribution.

2.1 Deﬁnitions and Basic Properties

Here we introduce some basic and related deﬁnitions and properties.

Deﬁnition 2.1.1. [17] Let X be a real-valued random variable with cumulative distribution F (x) and quantile function x(F ), and let X1:n ≤ X2:n ≤ ... ≤ Xn:n be the order statistics of a random sample of size n drawn from the distribution of X. Deﬁne the L-moments of X to be the quantities   Xr−1 −1 k r − 1 λr ≡ r (−1)   EXr−k:r r = 1, 2, .... (2.1.1) k=0 k

The L in “L-moments” emphasizes that λr is a linear function of the expected order statistics. Furthermore, as noted in [17], the natural estimator of λr based on an observed sample of data is a linear combination of the ordered data values. From Theorem 1.7.2, the (pdf) of the jth order statistic is given by:  

r j−1 r−j fj(x) = j   f(x)[F (x)] [1 − F (x)] j r! = [F (x)]j−1[1 − F (x)]r−jf(x). (j − 1)!(r − j)!

The expectation of an order statistic from eqn.(1.1.1) may be written as:

Z ∞ EXj:r = xfj(x) dx Z−∞ ∞ r! = x [F (x)]j−1[1 − F (x)]r−jf(x) dx. −∞ (j − 1)!(r − j)!

29 Hence, Z 1 r! j−1 r−j EXj:r = x[F (x)] [1 − F (x)] dF (x). (2.1.2) (j − 1)!(r − j)! 0 Lemma 2.1.1. [11] A ﬁnite mean implies ﬁnite expectation of all order statistics.

R 1 Proof. Assume that the mean µ = 0 x(u)du is ﬁnite. So, x(u) is integrable in the interval (0,1). Since from eqsn.(2.3.2) and (2.3.3) we have: Z 1 (j − 1)! (r − j)! uj−1[1 − u]r−jdu = B(j, r − j + 1) = is ﬁnite, 0 (r!) then uj−1[1−u]r−j is integrable in the interval (0,1). Hence, x(u) uj−1[1−u]r−j is integrable in the interval (0,1) (because the product of two integrable functions on any interval is an integrable function on this interval) and so

Z 1 x(u) uj−1[1 − u]r−jdu is ﬁnite. 0 From eqn.(2.1.2),

Z 1 r! j−1 r−j EXj:r = x(u) u [1 − u] du is finite· (j − 1)!(r − j)! 0 Therefore, a finite mean implies finite expectation of all order statistics.

Let’s rewrite the deﬁnition of the L-moment given in eqn.(2.1.1) to a simpler form that is easy in use. Change variable u = F (x). Let Q be the inverse of function F ; i.e., Q(F (x)) = x or F (Q(u)) = u:

Z 1 r! r−k−1 k EXr−k:r = Q(u) u (1 − u) du. (2.1.3) (j − 1)!(r − j)! 0 Substitute from eqn.(2.1.3) into eqn.(2.1.1) :   Z Xr−1 r − 1 r! 1 λ ≡ r−1 (−1)k   Q(u) ur−k−1(1 − u)k du. r (r − 1 − k)!k! k=0 k 0

For convenience, consider λr+1 instead of λr:

30   Z Xr r (r + 1)! 1 λ ≡ (r + 1)−1 (−1)k   Q(u) ur−k(1 − u)k du. r+1 (r − k)!k! k=0 k 0 Note that (r + 1)−1 (r + 1)! = r! and rearrange terms:

 2 Z 1 Xr k r r−k k λr+1 = (−1)   u (1 − u) Q(u) du. (2.1.4) 0 k=0 k Expand (1 − u)k in powers of u :

 2   Z 1 Xr Xk k r r−k k−j k k−j λr+1 = (−1)   u (−1)   u Q(u) du 0 k=0 k j=0 j     Z 2 1 Xr Xk r k = (−1)j     ur−j Q(u) du. 0 k=0 j=0 k j Interchange order of summation over j and k:

 2   Z 1 Xr Xr j r k r−j λr+1 = (−1)     u Q(u) du. 0 j=0 k=j k j Reverse order of summation: set m = r − j, n = r − k:

 2   Z 1 Xr Xm r−m r r − n m λr+1 = (−1)     u Q(u) du 0 m=0 n=0 r − n r − m

 2   Z 1 Xr n Xm o r−m r r − n m λr+1 = (−1)     u Q(u) du. (2.1.5) 0 m=0 n=0 r − n r − m

Note that  2         r r − n r r m     =       (2.1.6) r − n r − m n m n (expand the binomial coeﬃcients in terms of factorials) and that

31             Xm r m Xm r m r + m r + m     =     =   =   (2.1.7) n=0 n n n=0 r − n n r m (second equality follows because to choose r items from r + m we can choose from the ﬁrst m items and r − n from the remaining r items, for any n in 0, 1, ..., m). From (2.2.5) and (2.2.6), we have:

 2       Xm r r − n r r + m     =     , (2.1.8) n=0 r − n r − m m m and substituting into (2.1.5) gives     Z 1 Xr r−m r r + m m λr+1 = (−1)     u Q(u) du 0 m=0 m m     Z 1 Xr r−m r r + m m λr+1 = (−1)     x(F ) F dF. (2.1.9) 0 m=0 m m Let     r r + m ∗ r−m     pr,m = (−1) , (2.1.10) m m and Xr ∗ ∗ m Pr (F ) = pr,m F . (2.1.11) m=0 Substituting (2.1.11) into (2.1.9) we have [11]: Z 1 ∗ λr = x(F ) Pr−1(F ) dF, r = 1, 2,.... (2.1.12) 0

Example 2.1.1. To ﬁned λ2, substitute r = 2 in eqn.(2.1.1),   1 X1 1 λ = (−1)k   EX 2 2 2−k:2 k=0 k     h i 1 0 1 1 1 = (−1)   EX2:2 + (−1)   EX1:2 2 0 1 1h i 1 = EX − EX = E(X − X ). 2 2:2 1:2 2 2:2 1:2 32 And we can substitute r = 2 in eqn.(2.1.12),

Z 1 ∗ λ2 = x(F ) P1 (F ) dF 0 Z 1 h X1 i ∗ m = x(F ) p1,m F dF from eq.n.(2.1.11) 0 m=0 Z 1 h i ∗ 0 ∗ 1 = x(F ) p1,0F + p1,1F dF 0         Z 1 h 1 1 1 2 i = x(F ) (−1)1     + (−1)0     F dF from (2.1.10) 0 2 0 1 1 Z 1 = x(F )(2F − 1) dF. 0

Hence, Z 1 1 λ2 = E(X2:2 − X1:2) = x.(2F − 1) dF. 2 0 The ﬁrst few L-moments are:

Z 1 λ1 = EX = x. dF, Z0 1 1 λ2 = E(X2:2 − X1:2) = x.(2F − 1) dF, 2 0 Z 1 1 2 λ3 = E(X3:3 − X2:3 + X1:3) = x.(6F − 6F + 1) dF, 3 0 Z 1 1 3 2 λ4 = E(X4:4 − 3X3:4 + 3X2:4 − X1:4) = x.(20F − 30F + 12F − 1) dF. 4 0 The use of L-moments to describe probability distributions is justiﬁed by the next theorem. As shown in [17], λ2 is a measure of the scale or dispersion of the random variable X. It is often convenient to standardize the higher moments λr, r ≥ 3, so that they are independent of the units of measurement of X.

Deﬁnition 2.1.2. [18] Deﬁne the L-moment ratios of X to be the quantities

τr ≡ λr/λ2, r = 3, 4, ....

33 Note that [17]:

τ3 = λ3/λ2 is called L-skewness,

τ4 = λ4/λ2 is called L-kurtosis. It is also possible to deﬁne a function of L-moments which is analogous to the coeﬃcient of variation: this is the L − CV, τ ≡ λ2/λ1. Bounds on the numerical values of the L-moment ratios and L − CV is given by the following theorem.

Theorem 2.1.2. [18] Let X be a nondegenerate random variable with ﬁnite mean. Then the L-moment ratios of X satisfy | τr |< 1, r ≥ 3. If in addition X ≥ 0 almost surely, then τ, the L − CV of X, satisﬁes 0 < τ < 1.

Proof. Deﬁne Qr(t) by (−1)r dr t(1 − t)Q (t) = [t(1 − t)]r+1, r r! dtr (1,1) where Qr(t) is the Jacobi polynomial Pr (2t − 1). Then,

r r+1 d (−1) d r+1 [t(1 − t)Qr(t)] = r+1 [t(1 − t)] dt r! dt   (−1)r dr+1 Xr+1 r + 1 = (−1)k   tr+1+k r! dtr+1 k=0 k   (−1)r Xr+1 r + 1 dr+1 = (−1)k   [tr+1+k] r! dtr+1 k=0 k   (−1)r Xr+1 r + 1 ¡ ¢¡ ¢ ¡ ¢ = (−1)k   r + 1 + k r + k ..... k + 1 tk r! k=0 k Xr+1 (r + 1)!(r + 1 + k)! = −(r + 1) (−1)r−k+1 tk (r + 1)!k!(r + 1 − k)!k! k=0     Xr+1 r + 1 r + 1 + k = −(r + 1) (−1)r−k+1     tk k=0 k k Xr+1 r−k+1 ∗ k = −(r + 1) (−1) pr+1,kt k=0 ∗ ¡ ¢ = −(r + 1)Pr+1,k t .

34 Hence, d [t(1 − t)Q (t)] = −(r + 1)P ∗ (t). dt r r+1 Then, ¡ ¢ −1 d P ∗ t = [t(1 − t)Q (t)]. r+1 r + 1 dt r Therefore, ¡ ¢ −1 d P ∗ F = [F (1 − F )Q (F )]. r−1 r − 1 dF r−2

So, from eq.n.(2.1.12), Z −1 d h i λ = x(F ) F (1 − F )Q (F ) dF. r r − 1 dF r−2 Now, integrating by parts: Z −1 h i λ = x(F )F (1 − F )Q (F ) − F (1 − F )Q (F )dx r r − 1 r−2 r−2 h h i i Z h i −1 −1 = − xF (x) 1 − F (x) (r − 1) Qr−2(F (x)) + F (x) 1 − F (x) (r − 1) Qr−2(F (x))dx. h i Since xF (x) 1 − F (x) → 0 as x approaches the endpoint of the distribution, then Z h i −1 λr = F (x) 1 − F (x) (r − 1) Qr−2(F (x))dx. (2.1.13)

Since (−1)r 1 dr Q (t) = [t(1 − t)]r+1, r r! t(1 − t) dtr then Qt(0) = 1. In the case r = 2,

Z £ ¤ λ2 = F (x) 1 − F (x) dx. (2.1.14)

Now, 0 ≤ F (x) ≤ 1 for all x. So, Z −1 |λr| ≤ (r − 1) |F (1 − F )Qr−2(F )|dx Z −1 = (r − 1) |Qr−2|F (1 − F )dx Z −1 ≤ (r − 1) sup |Qr−2(t)|F (1 − F )dx 0≤t≤1 −1 = (r − 1) sup |Qr−2(t)|λ2. 0≤t≤1

35 We have (see [30])

sup |Qr(t)| = r + 1 0≤t≤1 with the supremume being attained only at t = 0 or t = 1 . Thus, (see [18]), |λr| ≤ λ2, with equality only if F (x) can take only the values 0 and 1; i.e., only if X is degenerate.

Thus, a nondegenerate distribution has |λr| ≤ λ2, which together with λ2 > 0 implies

|τr| < 1.

If X ≥ 0 almost surely, then λ1 = EX > 0 and λ2 > 0. So,

λ τ = 2 > 0. λ1

Furthermore, EX1:2 > 0. So,

τ − 1 = (λ2 − λ1)/λ1 = −EX1:2/λ1 < 0.

36 2.2 Probability Weighted Moments

Here we are about to have a tool by which we can easily ﬁnd the L-moments for any distribution.

Deﬁnition 2.2.1. [14] The probability weighted moments (PWMs) of a random variable X with a cumulative distribution function u = F (X) is the quantities

Z 1 © p r sª p r s Mp,r,s = E X F (X) (1 − F (X)) = X F (X) (1 − F (X)) dF r = 0, 1,.... 0 If we write a cumulative distribution function F (X) = u, then the quantile function is x(u) and

Z 1 © p r sª p r s Mp,r,s = E x(u) u (1 − u) = x(u) u (1 − u) du r = 0, 1,.... 0

A particular useful special cases are the probability weighted moments αr = M1,0,r and

βr = M1,r,0. For a distribution that has a quantile function x(u),

Z 1 r αr = x(u)(1 − u) du, 0

Z 1 r βr = x(u)u du. (2.2.1) 0 These equations may be contrasted with the deﬁnition of the ordinary moments, which may be written as Z 1 E(Xr) = {x(u)}r du. 0 Conventional moments involve successively higher powers of the quantile functions x(u), whereas probability weighted moments involve successively higher powers of u or 1 − u and may be regarded as integrals of x(u) weighted by the polynomials ur or (1 − u)r [15].

The probability weighted moments αr and βr have been used as the basis of methods for estimating parameters of probability distributions. However, they are diﬃcult to interpret directly as measures of the scale and shape of a probability distribution. This

37 information is carried in certain linear combinations of the probability weighted moments.

For example, estimates of scale parameters of distributions are multiples of α0 − 2α1 or

2β1 − β0. The skewness of a distribution can be measured by 6β2 − 6β1 + β0 ([15]).

L-moments are linear combination of probability-weighed moments [28], since

Z 1 Z 1 Xr ∗ ∗ m λr+1 = xPr (F )dF = x(F )pr,mF dF. 0 0 m=0 Xr Z 1 Xr ∗ m ∗ = pr,m x(F )F dF = pr,mβm. (2.2.2) m=0 0 m=0 From e.qn. (2.1.4), we have

 2 Z 1 Xr k r r−k k λr+1 = (−1)   u (1 − u) Q(u) du. (2.2.3) 0 k=0 k

Expand ur−k in powers of (1 − u):

 2   Z 1 Xr Xr−k k r k r−k−j r − k r−k−j λr+1 = (−1)   (1 − u) (−1)   (1 − u) Q(u) du 0 k=0 k j=0 j     Z 2 1 Xr Xr−k r r − k = (−1)r−j     (1 − u)r−j Q(u) du· 0 k=0 j=0 k j     Z 2 1 Xr Xr−k r r − k = (−1)r (−1)j     (1 − u)r−j Q(u) du· 0 k=0 j=0 r − k j     Z 2 1 Xr Xk r k = (−1)r (−1)j     (1 − u)r−j Q(u) du· 0 k=0 j=0 k j Interchange order of summation over j and k:

 2   Z 1 Xr Xr r j r k r−j λr+1 = (−1) (−1)     (1 − u) Q(u) du· 0 j=0 k=j k j

38 Reverse order of summation, set m = r − j, n = r − k:

 2   Z 1 Xr Xm r r−m r r − n m λr+1 = (−1) (−1)     (1 − u) Q(u) du 0 m=0 n=0 r − n r − m

 2   Z 1 Xr n Xm o r r−m r r − n m λr+1 = (−1) (−1)     (1 − u) Q(u) du. (2.2.4) 0 m=0 n=0 r − n r − m

Note that  2         r r − n r r m     =       (2.2.5) r − n r − m n m n (expand the binomial coeﬃcients in terms of factorials) and that

            Xm r m Xm r m r + m r + m     =     =   =   (2.2.6) n=0 n n n=0 r − n n r m

(second equality follows because to choose r items from r + m we can choose from the ﬁrst m items and r − n from the remaining r items, for any n in 0, 1, ..., m). From eq.n.(2.2.5) and eq.n.(2.2.6), we have:

 2       Xm r r − n r r + m     =     , (2.2.7) n=0 r − n r − m m m and substituting into 2.4.1 gives     Z 1 Xr r r−m r r + m m λr+1 = (−1) (−1)     (1 − u) Q(u) du 0 m=0 m m

    Z 1 Xr r r−m r r + m m λr+1 = (−1) (−1)     x(F ) (1 − F ) dF. (2.2.8) 0 m=0 m m

Xr Z 1 r ∗ m λr+1 = (−1) pr,m x(F ) (1 − F ) dF (2.2.9) m=0 0

39 Xr r ∗ λr+1 = (−1) pr,mαm. m=0 Hence, Xr Xr ∗ r ∗ λr+1 = pr,m βm = (−1) pr,m αm. (2.2.10) m=0 m=0 For example, the ﬁrst four L-moments are related to the PWMs as follows [25]:

λ1 = β0 = α0,

λ2 = 2β1 − β0 = α0 − 2α1, (2.2.11) λ3 = 6β2 − 6β1 + β0 = α0 − 6α1 + 6α2,

λ4 = 20β3 − 30β2 + 12β1 + β0 = α0 − 12α1 + 30α2 − 20α3.

2.3 Relation of L-moments with Order Statistic

From (1.7.1), the cdf of rth order statistic is given by:   Xn n k£ ¤n−k Fr(x) =   F (x) 1 − F (x) . (2.3.1) k=r k

Deﬁnition 2.3.1. ([10], Page 107) We deﬁne the Beta function B(a, b) as follows: Z 1 Γ(a)Γ(b) B(a, b) = ta−1(1 − t)b−1dt = · (2.3.2) 0 Γ(a + b) Note 2.3.1. If a, b are positive integers, then from Note 1.4.3 we can write (a − 1)!(b − 1)! B(a, b) = · (2.3.3) (a + b − 1)!

Deﬁnition 2.3.2. [25] The incomplete Beta function Ix(a, b) is deﬁned via the Beta function B(a,b) as follows:

Z x 1 a−1 b−1 Ix(a, b) = t (1 − t) dt. (2.3.4) B(a, b) 0 Theorem 2.3.2. The expression   Xn n k£ ¤n−k Fr(x) =   F (x) 1 − F (x) . k=r k

40 can be written in terms of an incomplete Beta function as:   Z F (x) n r−1 n−r Fr(x) = r   u (1 − u) du = IF (x)(r, n − r + 1). r 0 Proof. Claim:   Z Xn n Γ(a + b) x   xk(1 − x)n−k = ta−1(1 − t)b−1dt, Γ(a)Γ(b) k=a k 0 where n = a + b − 1 , Γ is the gamma function and 0 < x < 1. Proof of the claim: First, we want to ﬁnd a formula for Z x ta−1(1 − t)b−1dt. 0 Integrating by partes, let’s put u = (1 − t)b−1, dv = ta−1dt, then ta du = −(b − 1)(1 − t)b−2dt, v = · a So, Z x ¯ Z x ta(1 − t)b−1 ¯x ta ta−1(1 − t)b−1dt = ¯ + (b − 1)(1 − t)b−2dt. 0 a 0 0 a Hence, Z Z x xa(1 − x)b−1 (b − 1) x ta−1(1 − t)b−1dt = + ta(1 − t)b−2dt. (2.3.5) 0 a a 0 Now, by formula (2.3.5), we have: Z Z x xa(1 − x)b−1 (b − 1) x ta−1(1 − t)b−1dt = + ta(1 − t)b−2dt 0 a a 0 Z xa(1 − x)b−1 (b − 1)hxa+1(1 − x)b−2 (b − 2) x i = + + ta+1(1 − t)b−3dt a a a + 1 a + 1 0 Z xa(1 − x)b−1 (b − 1)xa+1(1 − x)b−2 (b − 1)(b − 2) x = + + ta+1(1 − t)b−3dt a a(a + 1) a(a + 1) 0

xa(1 − x)b−1 (b − 1)xa+1(1 − x)b−2 = + a a(a + 1)

Z (b − 1)(b − 2)hxa+2(1 − x)b−3 (b − 3) x i + + ta+2(1 − t)b−4dt a(a + 1) a + 2 a + 2 0

41 xa(1 − x)b−1 (b − 1)xa+1(1 − x)b−2 (b − 1)(b − 2)xa+2(1 − x)b−3 = + + a a(a + 1) a(a + 1)(a + 2)

Z (b − 1)(b − 2)(b − 3) x + ta+2(1 − t)b−4dt a(a + 1)(a + 2) 0

xa(1 − x)b−1 (b − 1)xa+1(1 − x)b−2 (b − 1)(b − 2)xa+2(1 − x)b−3 = + + a a(a + 1) a(a + 1)(a + 2)

Z (b − 1)(b − 2)(b − 3) x + ... + ta+2(1 − t)b−4dt. a(a + 1)(a + 2) 0 Therefore, Z Z Γ(a + b) x (a + b − 1)! x ta−1(1 − t)b−1dt = ta−1(1 − t)b−1dt (from Note 1.4.3) Γ(a)Γ(b) 0 (a − 1)!(b − 1)! 0

(a + b − 1)! hxa(1 − x)b−1 (b − 1)xa+1(1 − x)b−2 (b − 1)(b − 2)xa+2(1 − x)b−3 = + + (a − 1)!(b − 1)! a a(a + 1) a(a + 1)(a + 2)

Z (b − 1)(b − 2)(b − 3) x i + ... + ta+2(1 − t)b−4dt a(a + 1)(a + 2) 0

(a + b − 1)! xa(1 − x)b−1 (a + b − 1)! (b − 1)xa+1(1 − x)b−2 = × + × (a − 1)!(b − 1)! a (a − 1)!(b − 1)! a(a + 1)

(a + b − 1)! (b − 1)(b − 2)xa+2(1 − x)b−3 + × (a − 1)!(b − 1)! a(a + 1)(a + 2)

Z (a + b − 1)! (b − 1)(b − 2)(b − 3) x + ... + × ta+2(1 − t)b−4dt (a − 1)!(b − 1)! a(a + 1)(a + 2) 0

(a + b − 1)! xa (1 − x)b−1 (a + b − 1)! xa+1 (1 − x)b−2 = + a!(b − 1)! (a + 1)! (b − 2)!

Z (a + b − 1)! xa+2 (1 − x)b−3 (a + b − 1)! x + + ta+2(1 − t)b−4dt (a + 2)! (b − 3)! (b − 4)! (a + 2)! 0

42 (a + b − 1)! xa (1 − x)b−1 (a + b − 1)! xa+1 (1 − x)b−2 = + a!(b − 1)! (a + 1)! (b − 2)!

Z (a + b − 1)! xa+2 (1 − x)b−3 (a + b − 1)! x + + ... + ta+b−2(1 − t)b−bdt (a + 2)! (b − 3)! (a + b − 2)! (b − b)! 0

(a + b − 1)! xa (1 − x)b−1 (a + b − 1)! xa+1 (1 − x)b−2 = + a!(b − 1)! (a + 1)! (b − 2)!

(a + b − 1)! xa+2 (1 − x)b−3 (a + b − 1)! xa+b−1 (1 − x)0 + + ... + (a + 2)! (b − 3)! (a + b − 1)! 0!     a + b − 1 a + b − 1 =   xa(1 − x)b−1 +   xa+1(1 − x)b−2 a a + 1     a + b − 1 a + b − 1 +   xa+2(1 − x)b−3 + ... +   xa+b−1(1 − x)0 a + 2 a + b − 1   aX+b−1 a + b − 1 =   xk (1 − x)a+b−1−k k=a k   Xn n =   xk (1 − x)n−k, where n = a + b − 1. k=a k This completes the proof of the claim. Now, we want to show that   Z F (x) n r−1 n−r Fr(x) = r   u (1 − u) du = IF (x)(r, n − r + 1). r 0

From eqn.(2.3.4), we have

Z F (x) 1 r−1 n−r IF (x)(r, n − r + 1) = u (1 − u) du. (2.3.6) B(r, n − r + 1) 0 Indeed; note that: 1 Γ(n + 1) Γ(r + n − r + 1) = = . B(r, n − r + 1) Γ(r)Γ(n − r + 1) Γ(r)Γ(n − r + 1)

43 Substitute in eqn.(2.3.6),

Z F (x) Γ(r + n − r + 1) r−1 n−r IF (x)(r, n − r + 1) = u (1 − u) du. Γ(r)Γ(n − r + 1) 0 By the Claim we have:   Xn n k£ ¤n−k IF (x)(r, n − r + 1) =   F (x) 1 − F (x) . k=r k

From eq.n.(2.3.1), we have:

IF (x)(r, n − r + 1) = Fr(x).

We want to connect the cdf of the jth order statistic with the probability wight moments.

The probability density function of Xr:n is given by Theorem 1.7.2 as follows:  

n r−1 £ ¤n−r fr(x) = r   F (x) 1 − F (x) f(x). (2.3.7) r

Now, the expected value of rth order statistics can be obtained as

Z ∞ E[Xr:n] = xfr(x)dx. (2.3.8) −∞ Substituting from e.qn.(2.3.7) into (2.3.8) and introducing a transformation, u = F (x) or x = F −1(u), 0 ≤ u ≤ 1, leads to:   Z 1 n r−1 n−r E[Xr:n] = r   x(u)u (1 − u) du· (2.3.9) r 0 Note that, x(u) denotes the quantile function of a random variable. The expectation of the maximum and minimum of a sample of size n can be easily obtained from eq.n.(2.3.9) by setting r = n and r = 1, respectively as follows:

44 Z 1 n−1 E[Xn,n] = n x(u)u du, (2.3.10) 0 and Z 1 n−1 E[X1:n] = n x(u)(1 − u) du. (2.3.11) 0 The probability weighted moments (PWMs)of a random variable was formally deﬁned by : Z 1 i j k i j k Mi,j,k = E[x(u) u (1 − u) ] = x(u) u (1 − u) du. 0 The following two forms of PWMs are particularly simple and useful:

Z 1 k αk = M1,0,k = x(u)(1 − u) du (k = 0, 1, ...n) (2.3.12) 0 and Z 1 k βk = M1,k,0 = x(u)u du (k = 0, 1, ....n). (2.3.13) 0 Comparing eq.ns (2.3.12) and (2.3.13), with e.qns (2.3.10) and (2.3.11), it can be seen that αk and βk, respectively, are related to the expectation of the minimum and maximum in a sample of size k as follows: 1 α = E[X ], k k + 1 1:k+1

1 β = E[X ](k ≥ 1). (2.3.14) k k + 1 k+1:k+1 In fact, PWMs are the normalized expectations of maximum/minimum of k random observations. The normalization is done by the sample of size k itself. From eq.n.(2.3.10), we notice that:

E[Xn:n] = nβn−1.

From e.qn.(2.3.13), we have: Z ∞ n−1 βn−1 = xF (x)f(x)dx. −∞ So, Z ∞ n−1 E[Xn:n] = xnf(x)F (x)dx. −∞

45 2.4 Properties of L-moments

The L-moments λ1 and λ2, the L − CV, τ, and L-moment ratios τ3 and τ4 are most useful quantities for summarizing probability distributions. Their most important properties are the following (proofs are given in [17], [18]):

1. Existence. If the mean of the distribution exists, then all of the L-moments exist.

2. Uniqueness. If the mean of the distribution exists, then the L-moments uniquely deﬁne the distribution. That is; no two distinct distributions have the same L- moment. Properties 1 and 2 are proved in the next theorem.

Theorem 2.4.1. [1] (i) The L-moments λr, r = 1, 2,... of a real-valued random variable X exists if and only if X has a ﬁnite mean.

(ii) A distribution whose mean exists is characterized by its L-moments λr : r = 1, 2, . . ..

Proof. We know that a ﬁnite mean implies a ﬁnite expectation of all order statistics

(see Lemma 2.1.1). Since the L-moments λr, r = 1, 2,... are a linear functions of

the expected order statistics, then the L-moments λr, r = 1, 2,... exist.

Conversely, if the L-moments λr, r = 1, 2,... of a real-valued random variable X

exist, then the mean= λ1 exists. For part (ii), we ﬁrst show that a distribution is characterized by the set

{EXr:r, r = 1, 2,...}.

Let X and Y be random variables with cumulative distribution functions F and G and quantile functions x(u) and y(u), respectively. Let (X) R 1 r−1 (Y ) R 1 r−1 ξr ≡ EXr:r = r 0 x F (x) dF (x), ξr ≡ EYr:r = r 0 x G(x) dG(x). Then,

Z 1 (X) (X) r+1 r ξr+2 − ξr+1 = x(r + 2)u − (r + 1)u du 0

46 Z 1 = ur.u(1 − u) dx(u) integrating by parts, 0 Z 1 r = u . dzX (u), 0 where zX (u), deﬁned by dzX (u) = u(1−u) dx(u), is an increasing function on (0,1). (X) (Y ) If ξr = ξr , r = 1, 2,..., then

Z 1 Z 1 r r u . dzX (u) = u . dzY (u). 0 0

Thus, zX and zY are distributions which have the same moments on the ﬁnite interval

(0,1). Consequently, by Theorem 1.4.4, zX (u) = zY (u). Hence, dzX = dzY . That means, u(1 − u) dx(u) = u(1 − u) dy(u). Since u(1 − u) 6= 0, then dx(u) = dy(u). This implies that x(u) = y(u), and so F = G. Conversely, if F = G, then x(u) = y(u) and

Z 1 Z 1 (X) r−1 r−1 (Y ) ξr = EXr:r = r xF (x) dF (x) = r xG(x) dG(x) = EYr:r = ξr . 0 0 We have shown that a distribution with ﬁnite mean is characterized by the set

{ξr : r = 1, 2, ...}. Now, we want to show that a distribution with ﬁnite mean is characterized by its

L-moments λr : r = 1, 2, . . .. Recall eq.n. (2.1.12),

Z 1 ∗ λr = x(F )Pr−1(F )dF, r = 1, 2, ..., 0     Xr r r + k ∗ ∗ k ∗ r−k     Pr (F ) = pr,kF , pr,k = (−1) . k=0 k k Since Xr−1 ∗ ∗ k Pr−1(F ) = pr−1,kF , k=0 then,

47 Z 1 h Xr−1 i ∗ k λr = x(F ) pr−1,k F dF 0 k=0 Z 1 Xr−1 Xr−1 Z 1 ∗ k ∗ k = pr−1,k x(F ) F dF = pr−1,k x(F ) {F (x)} dF 0 k=0 k=0 0 Xr−1 Z 1 Xr Z 1 ∗ k ∗ k−1 = pr−1,k x(F ) {F (x)} dF = pr−1,k−1 x(F ) {F (x)} dF k=0 0 k=1 0 Xr Z 1 Xr ∗ −1 k−1 ∗ −1 = pr−1,k−1k k x(F ) {F (x)} dF = pr−1,k−1 k ξk· k=1 | 0 {z } k=1 ξk

From [18], we have: Xr (2k − 1) r!(r − 1)! ξ = λ · r (r − k)! (r − 1 + k)! k k=1

Thus, a given set of λr determines a unique set of {ξr : r = 1, 2, ...}, since a

distribution with ﬁnite mean is characterized by the set {ξr : r = 1, 2, ...}. Therefor,

a distribution whose mean exists is characterized by its L-moments λr : r = 1, 2, . . ..

Thus, a distribution may be speciﬁed by its L-moments even if some of its conventional moments do not exist([18]).

3. Terminology([15], Page 24)

• λ1 is the L-location or mean of the distribution.

• λ2 is the L-scale.

• τ is the L-CV

• τ3 is the L-skewness.

• τ4 is the L-kurtosis.

4. Numerical values

• λ1 can take any value, because λ1 = E(X) and X may be positive or negative.

48 • λ2 ≥ 0, because λ2 = E(X2:2 − X2:1) and X2:2 ≥ X2:1.

• For any distribution that takes only positive values, 0 ≤ τ < 1, this is proved in Theorem 2.1.2.

• L-moment ratios satisfy |τ| < 1 for all r ≥ 3. This is proved in Theorem 2.1.2.

• Tighter bounds can be found for individual τr quantities. For example, bounds

for τ4 given τ3 are 1 (5τ 2 − 1) ≤ τ < 1 ([15])· 4 3 4

• For a distribution that takes only positive values, bounds for τ3 given τ are

2τ − 1 ≤ τ3 < 1 ([18])·

5. Linear transformation. Let X and Y be random variables with L-moments λr and ∗ λr, respectively, and suppose that Y = aX + b, a > 0. Then, ∗ (I) λ1 = aλ1 + b ; ∗ (II) λ2 = aλ2 ; ∗ (III) τr = τr , r ≥ 3.

Proof. (I) Assume that X and Y are random variables with cumulative distribution functions F and G and quantile functions x(u) and y(u).

Let u = GY (y). Then, −1 y = GY (u) = y(u). (2.4.1)

Since

u = GY (y) = P (Y ≤ y) = P (aX + b ≤ y) ³ ´ ³ y − b´ = P aX ≤ y − b = P X ≤ (because a > 0) a ³y − b´ = F , X a

−1 y−b y−b then FX (u) = a · So, x(u) = a · Hence,

y = ax(u) + b. (2.4.2)

49 From eq.n.(2.4.1) and eq.n(2.4.2) we have:

y(u) = ax(u) + b. (2.4.3)

Then, from eq.n.(2.4.3) we have:

Z 1 Z 1 λ1(Y ) = y(u)du = [ax(u) + b]du 0 0 Z 1 = a x(u)du + b = aλ1(X) + b. 0

Z 1 (II) λ2(Y ) = y(u)(2u − 1)du 0 Z 1 = [ax(u) + b](2u − 1)du 0 Z 1 Z 1 = a x(u)(2u − 1)du + b (2u − 1)du 0 0 = aλ2(X).

(III) From eq.n.(2.1.12),

Z 1 ∗ λr(Y ) = y(F )Pr−1(F )d F, r = 3, 4 ... 0 Z 1 ∗ = (ax(F ) + b)Pr−1(F )d F 0 Z 1 Z 1 ∗ ∗ = a x(F )Pr−1(F )d F + b Pr−1(F ) dF, by eq.n(1.6.9) 0 0 = aλr(X).

Hence, for all r ≥ 3,

τr(Y ) = λr(Y )/λ2(Y )

= aλr(X)/aλ2(X)

= τr(X).

50 6. Symmetry. Let X be a symmetric random variable with mean µ. That is; P (X ≥ µ + x) = P (X ≤ µ − x) for all x. Then, all of the odd-order L-moment ratios of X

are zero. That is; τr = 0, r = 3, 5, ....([15], Page 24).

Proof. Assume X has the cumulative distribution function FX (x) and the quantile function x(u). Claim: If X is a symmetric random variable with mean µ then,

x(u) = −x(1 − u).

Proof of the claim: Conceder, X is a symmetric random variable with mean µ, that means, P (X ≥ µ + x) = P (X ≤ µ − x), then P (X ≥ x) = P (X ≤ −x).

Now, FX (−x) = P (X ≤ −x) = P (X ≥ x) = 1 − P (X ≤ x) = 1 − FX (x). −1 −1 −1 Hence, −x(u) = FX [FX (−x)] = FX [1 − FX (x)] = FX (1 − u) = x(1 − u). Therefor, x(u) = −x(1 − u). This completes the proof of the claim.

Now we want to calculate λ2r+1, r = 1, 2, 3 ... for a symmetric random variable X,

then we want to ﬁned it’s τ2r+1, r = 1, 2, 3 ... Replace 2r + 1 with r in eqn.(2.1.12) we have

Z 1 Z 1 X2r ∗ ∗ m λ2r+1 = x(u) P2r(u) du = p2r,m x(u) u du. (2.4.4) 0 0 m=0 And replace 2r + 1 with r + 1 in eqn.(2.2.9) we have

X2r Z 1 2r+1 ∗ m λ2r+1 = (−1) p2r,m x(u) (1 − u) du m=0 0 Z 1 X2r ∗ m = − p2r,m x(u) (1 − u) du 0 m=0 Z 1 X2r ∗ m = − p2r,m − x(1 − u) (1 − u) du, by the claim 0 m=0 Z 1 X2r ∗ m = − p2r,m x(1 − u) (1 − u) d(1 − u) 0 m=0

Z 1 X2r ∗ m = − p2r,m x(z) z dz, where z = 1 − u. (2.4.5) 0 m=0

51 From eqn’s (2.4.4) and (2.4.5) we have λ2r+1 = −λ2r+1, that means, λ2r+1 = 0 for all r = 1, 2, 3,.... Then, λ2r+1 0 τ2r+1 = = = 0. λ2 λ2

2.5 L-skewness and L-kurtosis

The main features of a probability distribution should be well-summarized by the following four measures: the mean or L-location (λ1), the L-scale λ2, the L-skewness τ3 and the

L-kurtosis τ4. We now consider these measures, particulary τ3 and τ4 in more details.

The L-moment measure of location is the mean, λ1. This is a well-established and familiar quantity which needs no further description or justiﬁcation here [17].

The L-scale λ2 is also long established in statistic, for it is, apart from a scalar multiple, , the expectation of Gini s mean diﬀerence statistic. To compare λ2 with the more familiar scale measure σ, the standard deviation, write

1 1 λ = E(X − X ), σ = E(X − X )2 2 2 2:2 1:2 2 2 2:2 1:2

Both quantities measure the diﬀerence between two randomly drawn elements of a distribution, but σ2 gives relatively more weight to the largest diﬀerence [17].

λk λ2 is used to obtain scale-free higher-order descriptive measure, τk = , k ≥ 3 called λ2 L-moment ratios, very conveniently for practical use and interpretation.

Table 2.1 shows the L-skewness for some common distributions. The L-skewness τ3 is a dimensionless analogue of λ3. By theorem 2.4.1, τ3 takes values between −1 and +1.

EX3:3 − 2EX2:3 + EX1:3 τ3 = EX3:3 − EX1:3 shows that τ3 is similar in form to a measure of skewness.

52 Table 2.1: L-skewness of some common distribution Distribution L-skewness

Uniform 0

1 Exponential 3 Gumble 0.1699 Logistic 0 Normal 0

Generalized Pareto τ3 = (1 − k)/(3 + k) Generalized extreme value 2(1 − 3−k)/(1 − 2−k) − 3 Generalized logistic −k

Table 2.2: L-kurtosis of some common distributions Distribution L-kurtosis

Uniform 0

1 Exponintial 6 Gumble 0.1504

1 Logistic 6 Normal 0.1226 Generalized Pareto (1 − k)(2 − k)/(3 + k)(4 + k) Generalized extreme value (1 − 6.2−k + 10.3−k − 5.4−k)/(1 − 2−k) Generalized logistic (1 + 5k2)/6

53 L-kurtosis, τ4, is equally diﬃcult to interpret uniquely and is best thought of as a measure similar to κ but giving less weight to the extreme tails of the distribution [17]. Table 2.2 shows the L-kurtosis for some common distributions:

2.6 L-moments of a Polynomial Function of Random Variables

In this section, we ﬁnd the kth PWMs of a random variable Y = Xm and we apply this relation to the standard normal distribution and to the exponential distribution.

The kth PWMs of a random variable X with quantile function x(u) is given from eq.n.(2.2.1) by: Z 1 k βk = x(u)u du. 0 The quantile function of the random variable Y = Xm follows from a transformation y(u) = xm(u): Since

m 1 1 FY (y) = P (Y ≤ y) = P (X ≤ y) = P (X ≤ y m ) = FX (y m )

1 −1 −1 m FY (u) = y, FX (u) = y .

Then, −1 −1 m m y(u) = FY (u) = y = (FX (u)) = x (u).

Therefore, the kth PWMs of Xm is given by:

Z 1 −1 m k βk = [FX (u)] u du. 0 In particular, if X is a standard normally distributed variable, then the following PWM,s is: Z 1 −1 m k βk = [Φ (u)] u du 0 and can be calculated numerically as shown in Table 2.3.

54 R 1 −1 m k Table 2.3: [25] Matrix B with numerical evaluations of βk = 0 (Φ (u)) u du X0 X1 X2 X3 X4 X5

β0 1 0 1 0 3 0

β1 1/2 0.282 0.5 0.705 1.5 3.032

β2 1/3 0.282 0.425 0.705 1.400 3.032

β3 1/4 0.257 0.388 0.675 1.350 2.969

β4 1/5 0.233 0.360 0.650 1.305 2.907

β5 1/6 0.211 0.337 0.618 1.266 2.848

In particular, if X is exponentially distributed, then the following PWM,s can be written as: Z 1 m k βk = [ξ − α ln(1 − u)] u du 0 and can be calculated numerically as shown in Table 2.4. L-moments are linear combinations of the PWMs, from eq.n.s(2.2.11)

λ1 = β0,

λ2 = −β0 + 2β1,

λ3 = β0 − 6β1 + 6β2,

λ4 = −β0 + 12β1 − 30β2 + 20β3.

Then, L-moments are given by the matrix multiplication λ = AB, in which       λ 1 0 0 0 β  1     0         λ   −1 2 0 0   β   2     1  λ =   ,A =   ,B =   .  λ   1 −6 6 0   β   3     2  λ4 −1 12 −30 20 β3

Furthermore, L-moments are linear combinations of observations and therefore the L- moments of the summation of two random variables is given by the summation of the L-moments of the random variables separately[25].

55 R 1 m k Table 2.4: [25] Matrix B with numerical evaluation of βk = 0 (ξ − α log(1 − u)) u du X0 X1 X2 X3 X4 2 2 3 3 2 2 2 3 2 2 β0 1 ξ + α ξ + 2α + 2ξα ξ + 6α + 6ξα + 3ξ + 3ξ α 4ξ α + 12ξ α + 24ξα3 + ξ4 + 24α4 2 3 4 4 3 β1 1/2 1/2ξ + 3/4α 3/2ξα+ 21/4ξα + 1/2ξ 1/2ξ + 93/4α + 3ξ α 7/4ξ2 + 1/2ξ2 +45/8α3 + 9/4ξ2α +21/2ξ2α2 + 45/2ξα3 2 2 3 3 4 β2 1/3 1/3ξ + 11/18α 11/9ξα + 85/54α 11/6ξ α + 575/108α + 575/27ξα + 3661/162α + +1/3ξ2 1/3ξ3 + 85/18ξα2 22/9ξ3α + 85/9ξ2α2 + 1/3ξ4 2 3 2 2 2 3 β3 1/4 1/4ξ + 25/48α 25/24ξα + 415/288α 1/4ξ + 25/16ξ α + 5845/ 415/48ξ α + 5845/288ξα + +1/4ξ2 1152α3 + 415/96ξα2 76111/3456α4 + 25/12ξ3α+ 1/4ξ4 2 3 2 3 2 2 β4 1/5 1/2ξ + 137/300α 1/5ξ + 137/ 1/5ξ + 137/100ξ α + 12019/ 137/75ξ α + 12019/1500ξ α 150ξα + 12019/ 3000ξα2 + 874853/180000α3 +874853/45000ξα3 + 1/5ξ4+ 9000α2 58067611/2700000α4 18000ξ2α3 + 3673451957/ 2 3 4 4 β5 1/6 1/2ξ + 49/120α 49/60ξα + 13489/ 49/40ξ α + 1/6ξ + 13489/ 1/6ξ + 68165041/3240000α + 10800α2 + 1/6ξ2 3600ξα2 + 336581/72000α3 49/30ξ3α + 13489/1800ξ2α2 +336581/18000ξα3

X5 4 4 3 2 β0 120ξα + 5ξ + 20ξ α + 60ξ2α3 + 120α5 + ξ5 4 4 β1 465/4ξα + 15/4ξ α+ 2ξ3α2 + 225/4ξ2α3+ 945/8α5 + 1/2ξ5 4 β2 18305/162ξα α+ 425/27ξ3α2 + 2875/54ξ2α3 +113155/972ξ5 + 1/3ξ5 4 4 β3 3805555/3456ξα + 125/48ξ α+ 2075/144ξ3α2 + 29225/576ξ2α3+ 4762625/41472α5 + 1/4ξ5 4 β4 58067611/540000ξα + 137/ 60ξ4α + 12019/900ξ3α2 + 874853/ 18000ξ2α3 + 3673451957/ 32400000α5 + 1/5ξ5 4 β5 68165041/648000ξα + 49/ 24ξ4α + 13489/1080ξ3α2 + 336581/ 7200ξ2α3 + 483900263/ 432000α5 + 1/6ξ5

56 2.7 Approximating a Quantile Function

In this section we introduce a theorem of a special importance. From this theorem that approximates a quantile function x(F ), we can find any distribution when we know it’s L-moments and this is by finding its quantile function x(F ). After that, we can find the cumulative distribution function F (x) to this distribution and then, we can get its probability distribution function f(x).

Theorem 2.7.1. [27]. Let X be a real-valued continuous random variable with ﬁnite variance, quantile function x(F ) and L-moment λr, r ≥ 1. Then the representation

X∞ ∗ x(F ) = (2r − 1)λrPr−1(F ), 0 < F < 1, r=1 is convergent in mean square, i.e., Xs ∗ Rs(F ) ≡ x(F ) − (2r − 1)λrPr−1(F ), r=1 the remainder after stopping the inﬁnite sum after s terms, satisﬁes

Z 1 2 {Rs(F )} dF → 0 as s → ∞ . 0 ∗ Proof. The shifted Legendre polynomials Pr−1(F ) are a natural choice as the basis of the approximation because they are orthogonal on 0 < F < 1 with constant weight function

∗ r(F ) = 1. We can say that {Pr−1(F ): r = 1, 2, 3 ...} is the set of eigenfunctions, and x(F ) is a function on [0,1], (see section 1.6). Now, we can apply e.qn. (1.6.4) in Corollary 1.6.2 to ﬁned x(F ) as follows:

X∞ ∗ x(F ) = cr Pr−1(F ), r=0

Z 1 1 ∗ where cr = r(F ) u(F ) Pr−1(F ) dF, µr 0 Z 1 Z 1 ∗ 2 ∗ 2 µr = r(F ) {Pr−1(F )} dF = {Pr−1(F )} dF. 0 0

57 Since r(F ) = 1 and by eq.n.(1.6.8),

Z 1 ∗ 2 1 {Pr (F )} dF = , 0 2r + 1 then, Z 1 ∗ 2 1 1 µr = {Pr−1(F )} dF = = · 0 2(r − 1) + 1 2r − 1 So, Z ³ ´−1 1 1 ∗ cr = u(F ) Pr−1(F ) dF = (2r − 1)λr. 2r − 1 0 Hence,

X∞ ∗ x(F ) = (2r − 1)λr Pr−1(F ). (2.7.1) r=0 Now,

Z Z s 1 1 n X o2 2 ∗ {Rs(F )} dF = x(F ) − (2r − 1)λrPr−1(F ) dF. 0 0 r=1

By e.qn.(2.7.1) and as s → ∞,

Z Z ∞ 1 1 n X o2 2 ∗ {Rs(F )} dF = x(F ) − (2r − 1)λrPr−1(F ) dF = 0· 0 0 r=1

Example 2.7.1. We can apply Theorem 2.7.1 to ﬁnd the quantile function x(F ) of the uniform distribution from it’s L-moments. Since, from subsection 3.9.1, the L-moments

1 1 for the uniform distribution are: λ1 = 2 (α+β), λ2 = 6 (β−α) and λr = 0, r = 3, 4, 5 ..., then, X∞ ∗ x(F ) = (2r − 1)λrPr−1(F ) r=1 X2 ∗ = (2r − 1)λrPr−1(F )(because λr = 0, for all r > 2) r=1

∗ ∗ = λ1P0 (F ) + 3λ2P1 (F )

58 1 1 = (α + β) + 3 × (β − α)[2F − 1] 2 6 1 1 = (α + β) − (β − α) + (β − α)F 2 2 1 = (α + β − β + α) + (β − α)F 2 = α + (β − α)F ·

Since the cumulative distribution function F (x) is the inverse function of quantile function x(F ), then we can ﬁnd the cumulative distribution function of the uniform distribution

F (x) by   0, x < α;  F (x) = (x − α)/(β − α), α ≤ x < β;    1, x ≥ β. and we can get the probability distribution function of the uniform distribution f(x) by f(x) = F 0(x) = 1/(β − α).

2.8 L-moments as Measures of Distributional Shape

In [12], Oja has deﬁned intuitively reasonable criteria for one probability distribution on the real line to be located further to the right (more dispersed, more skew, kurtotic) than another. Areal-valued function of a distribution that preserves the partial ordering of distributions implied by these criteria may then reasonably be called a “measure of location” (dispersion, skewness, kurtosis). The following theorem shows that τ3 and τ4 are, by Oja’s criteria, measures of skewness and kurtosis respectively.

Deﬁnition 2.8.1. [24] Let S ⊂ Rn be a nonempty convex set. Function f : S → R is said to be convex on S if for any x1, x2 ²S and all 0 ≤ α ≤ 1, we have

f(αx1 + (1 − α)x2) ≤ αf(x1) + (1 − α)f(x2).

Theorem 2.8.1. [12] Let X and Y be continues real-valued random variables with cumulative distribution functions F and G respectively, and L-moment λr(X) and λr(Y )

59 respectively.

(i) If Y = aX + b, then λ1(Y ) = aλ1(X) + b, λ2(Y ) = |a|λ2(X) , τ3(Y ) = τ3(X),

τ4(Y ) = τ4(X). −1 (ii) Let M (x) = G (F (x)) − x. If M (x) ≥ 0 for all x, then λ1(Y ) ≥ λ1(X). If M (x) is convex, then τ3(Y ) ≥ τ3(X).

Proof. (i) Since λ1(Y ) = E(Y ) = E(aX + b) = aE(X) + b = aλ1(X) + b and since Y = (aX + b), then

FY (y) = P (Y ≤ y) = P (aX + b ≤ y) = P (aX ≤ y − b), we have two cases: y−b y−b case(1): a > 0, then a = |a| and FY (y) = P (X ≤ |a| ) = FX ( |a| ), y−b y−b case(2): a < 0, then a = −|a| and FY (y) = P (−X ≤ |a| ) = P (X ≥ − |a| ) y−b y−b = P (X ≤ |a| ) = FX ( |a| ).

y−b Let u = FY (y) = FX ( |a| ). Then,

−1 y = FY (u) (2.8.1)

y−b −1 and |a| = FX (u). Now,

−1 y = |a|FX (u) + b. (2.8.2)

−1 −1 From eq.n.(2.8.1) and eq.n.(2.8.2), we have: FY (u) = |a|FX (u) + b. So, y(u) = |a|x(u) + b. (2.8.3)

Therefore,

Z 1 λ2(Y ) = y(u)(2u − 1)du 0 Z 1 = [|a|x(u) + b](2u − 1)du from eq.n.(2.8.3) 0 Z 1 Z 1 = |a| x(u)(2u − 1)du + b (2u − 1)du 0 0 ¯1 (X) 2 ¯ = |a|λ2 + b(u − u)¯ = |a|λ2(X) · 0

60 Claim: Xr−1 p∗ λ (Y ) = |a|λ (X) + b r−1,k · r r k + 1 k=0 Proof of the claim: From eq.n.(2.1.12),

Z 1 ∗ λr(Y ) = y(u)Pr−1(u)du 0 Z 1 ∗ = [|a|x(u) + b]Pr−1(u)du from eq.n.(2.8.3) 0 Z 1 Z 1 ∗ ∗ = |a| x(u)Pr−1(u)du + b Pr−1(u)du 0 0 Z 1 h Xr−1 i ∗ k = |a|λr(X) + b pr−1,ku du 0 k=0 Xr−1 h Z 1 i ∗ k = |a|λr(X) + b pr−1,k u du k=0 0 Xr−1 p∗ = |a|λ (X) + b r−1,k · r k + 1 k=0

So, from the claim, we have:

X2 p∗ λ (Y ) = |a|λ (X) + b 2,k 3 3 k + 1 k=0 h ∗ ∗ ∗ i p2,0 p2,1 p2,2 = |a|λ3(X) + b + + 1 2 3      h³ ´2 2 2 1³ ´1 2 3 = |a|λ3(X) + b − 1     + − 1     0 0 2 1 1     1³ ´0 2 4 i + − 1     3 2 2 h 1 1 i = |a|λ3(X) + b 1 − × 2 × 3 + × 1 × 6 h 2 i 3 = |a|λ3(X) + b 1 − 3 + 2 = |a|λ3(X).

61 Therefore,

τ3(Y ) = λ3(Y )/λ2(Y ) = |a|λ3(X)/|a|λ2(X)

= λ3(X)/λ2(X) = τ3(X), and

X3 p∗ λ (Y ) = |a|λ (X) + b 3,k 4 4 k + 1 k=0 h ∗ ∗ ∗ ∗ i p3,0 p3,1 p3,2 p3,3 = |a|λ4(X) + b + + + 1 2 3 4     h³ ´3 3 3 1³ ´2 3 4 = |a|λ4(X) + b − 1     + − 1     0 0 2 1 1         1³ ´1 3 5 1³ ´0 3 6 i + − 1     + − 1     3 2 2 4 3 3 h 1 1 1 i = |a|λ4(X) + b − 1 + × 3 × 4 − × 3 × 10 + × 1 × 20 h 2 i 3 4 = |a|λ4(X) + b − 1 + 6 − 10 + 5 = |a|λ4(X)·

Now,

τ4(Y ) = λ4(Y )/λ2(Y ) = |a|λ4(X)/|a|λ2(X)

= λ4(X)/λ2(X) = τ4(X)·

−1 To show that λ1(Y ) ≥ λ1(X), let y = G (F (x)). Since M (x) ≥ 0 for all x, then G−1(F ) ≥ x(F ) for all F . Hence, y(u) ≥ x(u) for all u. Therefore,

Z 1 Z 1 y(u)du ≥ x(u)du. 0 0

From eq.n.(2.1.12), λ1(Y ) ≥ λ1(X)·

For τ3, we want to show that τ3(Y ) ≥ τ3(X). Assume that the probability density functions of X and Y are respectively, f and g and let r(x) = f(x)/g{G−1(F (x))}. Because

62 M (x) is convex, r(x) = d M (x)/dx + 1 is increasing. Since y = G−1(F (x)),G(y) = F (x). This implies that dG(y) dF (x) dy = , and so, g(y) = f(x). dx dx dx So,

h i h i dy = f(x)/g(y) dx = f(x)/g{G−1(F (x))} dx = r(x) dx.

Now, by eq.n.(2.1.14), and from F (x) = G(y), dy = r(x) dx we have:

Z ∞ Z ∞ λ2(Y ) = G(y){1 − G(y)} dy = F (x){1 − F (x)}r(x)dx. −∞ −∞ Similarly,

Z ∞ Z ∞ λ3(Y ) = G(y){1 − G(y)}{2G(y) − 1} dy = F (x){1 − F (x)}{2F (x) − 1}r(x) dx. −∞ −∞

Thus, λ2(X)λ2(Y )τ3(Y ) − τ3(X) = λ3(Y )λ2(X) − λ3(X)λ2(Y ) which can be written as: Z ∞ Z ∞ F (x){1 − F (x)}{2F (x) − 1}r(x)dx. F (x){1 − F (x)}dx −∞ −∞ Z ∞ Z ∞ − F (x){1 − F (x)}{2F (x) − 1}. F (x){1 − F (x)}r(x) (2.8.4) −∞ −∞ wherein F (x){1−F (x)} is a positive function of x and 2F (x)−1 and r(x) are increasing. Chebyshev’s Other Inequality for integrals (see Theorem 1.6.6) implies that (2.8.4) is positive. Because λ2(X)λ2(Y ) > 0, it follows that τ3(Y ) ≥ τ3(X).

2.9 L-moments for some Distributions

In this section, we find the first four L-moments for some distributions and this will be used in chapter three in estimating the parameters of some distributions. This section is divided into four subsections: L-moments for uniform distribution, L- moments for exponential distribution, L-moments for logistic distribution and L-moments for generalized pareto distribution. In Table 2.5, we introduce the first four L-moments for some distributions.

63 Table 2.5: [17] L-moments of some common distributions Distribution F (x) or x(F ) L-moments 1 1 Uniform x = α + (β − α)F λ1 = 2 (β + α), λ2 = 6 (β − α), τ3 = 0, τ4 = 0 1 1 1 Exponential x = ξ − α log(1 − F ) λ1 = ξ + α, λ2 = 2 α, τ3 = 3 , τ4 = 6

Gumble x = ξ − α log(−logF ) λ1 = ξ + γα, λ2 = α log 2, τ3 = 0.1699, τ4 = 0.1504 Logistic x = ξ − α log{F/(1 − F )} λ = ξ, λ = α, τ = 0, τ = 1 1 2 3 4 6 x−µ 1 − 2 Normal F = Φ σ λ1 = µ, λ2 = π σ, τ3 = 0, τ4 = 0.1226 k Generalized x = ξ + α{1 − (1 − F ) }/k λ1 = ξ + α{1 − Γ(1 + k)}/k, λ2 = α/(1 + k)(2k + 1),

Pareto τ3 = (1 − k)/(3 + k), τ4 = (1 − k)(2 − k)/(3 + k)(4 + k) k −k Generalized x = ξ + α{1 − (− log F ) }/k λ1 = ξ + α{1 − Γ(1 + k)}/k, λ2 = α(1 − 2 )Γ(1 + k)/k, −k −k extreme value τ3 = 2(1 − 3 )/(1 − 2 ) − 3, −k −k −k −k τ4 = (1 − 6.2 + 10.3 − 5.4 )/(1 − 2 ) k Generalized x = ξ + α[1 − {(1 − F )/F } ]/k λ1 = ξ + α{1 − Γ(1 + k)Γ(1 − k)}/k, λ2 = αΓ(1 + k)(1 − k), logistic τ = −k, τ = (1 + 5k2)/6 3 4 log(x−ξ)−µ 2 2 Long-normal F = Φ σ λ1 = ξ + exp(η + σ /2), λ2 = exp(η + σ /2)erf(σ/2), R √ −1/2 σ/2 2 τ3 = 6π erf(x/ 3)exp(−x )dx/erf(σ/2) R 0 −α x α−1 −1/2 1 Gamma F = β 0 t exp(−t/β) dt/Γ(α) λ1 = αβ, λ2 = π βΓ(α + 2 )/Γ(α), τ3 = 6I1/3(α, 2α) − 3

2.9.1 L-moments for Uniform Distribution

In this subsection, we ﬁnd the L-moments for the uniform distribution. The uniform distribution has the probability density function([15], Page 191):

f(x) = 1/(β − α), and has the quantile function[17]:

x(F ) = α + (β − α)F.

We are about to find the first four L-moments of the uniform distribution. Before doing so, we have to determine the first for PWMs of the the uniform distribution. Z 1 r βr = x(F )F dF, r = 0, 1, 2,... 0 Z 1 h i Z 1 Z 1 = α + (β − α)F F rdF = αF rdF + (β − α)F r+1dF 0 0 0 ¯ ¯ F r+1 ¯1 (β − α) ¯1 α β − α = α ¯ + F r+2¯ = + · r + 1 0 r + 2 0 r + 1 r + 2 Then, α β − α β = + (2.9.1) r r + 1 r + 2

64 and,

β − α 1 λ = β = α + = (β − α) 1 0 2 2 hα β − αi α + β 1 λ = 2β − β = 2 + − = (β − α) 2 1 0 2 3 2 6 hα β − αi hα β − αi α + β λ = 6β − 6β + β = 6 + − 6 + + = 0 3 2 1 0 3 4 2 3 2

λ4 = 20β3 − 30β2 + 12β1 − β0 hα β − αi hα β − αi hα β − αi hα + β i = 20 + − 30 + + 12 + − = 0 · 4 5 3 4 2 3 2

Hence,

τ3 = λ3/λ2 = 0

τ4 = λ4/λ2 = 0 ·

Now, we write Note 2.9.1 and Note 2.9.2 because it is used in ﬁnd the others L-moments of the uniform distribution.

∗ Pr pr,m Proposition 2.9.1. m=0 m+1 = 0, for all r ≥ 1. R 1 ∗ Proof. From eq.n.(1.6.9) we have: 0 Pr (F )dF = 0, for all r ≥ 1. Since

Z 1 Z 1 Xr ∗ ∗ m Pr (F )dF = pr,mF dF, from eq.n.(2.1.11) 0 0 m=0 Xr Z 1 ∗ m = pr,m F dF m=0 0 r X m+1 ¯1 ∗ F ¯ = pr,m ¯ m + 1 0 m=0 Xr p∗ = r,m · m + 1 m=0 Therefor, Xr p∗ r,m = 0, for all r ≥ 1. (2.9.2) m + 1 m=0

65 ∗ Pr pr,m Proposition 2.9.2. m=0 m+2 = 0, for all r ≥ 2.

∗ ∗ Proof. Since P1 (F ) is orthogonal with Pr (F ) for all r ≥ 2 on the interval (0.1), then, for R 1 ∗ ∗ all r ≥ 2 we have: 0 P1 (F )Pr (F )dF = 0. Now,

Z 1 Z 1 h X1 i ∗ ∗ ∗ m ∗ P1 (F )Pr (F )dF = pr,mF Pr (F )dF, from eq.n.(2.1.11) 0 0 m=0 Z 1 h i ∗ ∗ ∗ = p1,0 + p1,1F Pr (F )dF 0 Z 1 h i ∗ = 2F − 1 Pr (F )dF, from eq.n.(2.1.10) 0

Z 1 Z 1 ∗ ∗ = 2 FPr (F )dF − Pr (F )dF 0 0 Z 1 ∗ = 2 FPr (F )dF 0 Z 1 ³ Xr ´ ∗ m = 2 F pr,mF dF, from eq.n.(2.1.11) 0 m=0 Xr Z 1 ∗ m+1 = 2 pr,m F dF m=0 0 r X m+2 ¯1 ∗ F ¯ = 2 pr,m ¯ m + 2 0 m=0 Xr p∗ = 2 r,m · m + 2 m=0 Hence, Xr p∗ r,m = 0 for all r ≥ 2. (2.9.3) m + 2 m=0

We are about to ﬁnd the others L-moments for the uniform distribution.

66 From eq.n.(2.2.2) and for all r ≥ 2 we have:

Xr ∗ λr+1 = pr,mβm m=0 Xr h α β − α i = p∗ + , from eq.n.(2.9.1) r,m m + 1 m + 2 m=0 Xr p∗ Xr p∗ = α r,m + (β − α) r,m m + 1 m + 2 m=0 m=0 = 0, from eqs.n.(2.9.2), (2.9.3).

That means, the L-moments for the uniform distribution, λr = 0 for all r ≥ 3 .

2.9.2 L-moments for Exponential Distribution

In this subsection, we ﬁnd the ﬁrst four L-moments for the exponential distribution. The exponential distribution has the cumulative distribution function([15], Page 192):

F (x) = 1 − exp{−(x − ξ)/α}, where ξ ≤ x < ∞.

Firstly we want to ﬁnd the quantile function of the exponential distribution. So, replace x(F ) with x, and F with F (x) we have: F = 1 − exp{−(x(F ) − ξ)/α}, then 1 − F = exp{−(x(F ) − ξ)/α}, hence, ln(1 − F ) = −(x(F ) − ξ)/α. Therefor, x(F ) = ξ − α ln(1 − F ). secondly, we want to ﬁnd the rth PWM for the exponential distribution:

Z 1 r βr = x(F ) F d F r = 0, 1, 2, ...... Z0 Z 1 h i ξ 1 = ξ − α ln(1 − F ) F rdF = − α F r ln(1 − F )d F · 0 r + 1 0 Now, we will ﬁnd Z 1 F r ln(1 − F )d F. 0 Integrating by parts, we get:

Z 1 ¯ Z 1 F r+1 ¯1 1 F r+1 F r ln(1 − F )d F = ln(1 − F )¯ + d F 0 r + 1 0 r + 1 0 1 − F

67 Z 1 1 F r+1 = d F · (2.9.4) r + 1 0 1 − F Let z = 1 − F . Then, dz = −dF , F = 1 − z.

So,   Xr+1 r + 1 F r+1 = (1 − z)r+1 = (−1)k   zk. k=0 k Therefore,     F r+1 Xr+1 r + 1 1 Xr+1 r + 1 = (−1)k   zk−1 = +   zk−1. 1 − F z k=0 k k=1 k

From eq.n (2.9.4),

  Z Z 1 −1 1 h1 Xr+1 r + 1 i F r ln(1 − F )dF = + (−1)k   zk−1 dz r + 1 z 0 0 k=1 k   h r+1 i¯ −1 X 1 r + 1 ¯0 = ln z + (−1)k   zk ¯ r + 1 k 1 k=1 k   1 h Xr+1 1 r + 1 i = (−1)k   r + 1 k k=1 k   1 Xr+1 1 r + 1 = (−1)k   . (2.9.5) r + 1 k k=1 k Hence,   ξ α Xr+1 1 r + 1 β = − (−1)k   r r + 1 r + 1 k k=1 k

β0 = ξ − α(−1) = ξ + α.

68 So,

λ = β = ξ + α 1 0       ξ α X2 1 2 ξ αh 2 1 2 i β = − (−1)k   = − −   +   1 2 2 k 2 2 2 k=1 k 1 2 ξ 3α = + · 2 4 Therefore, ³ξ 3α´ ³ ´ α λ2 = 2β1 − β0 = 2 + − ξ + α = 2 4  2      ξ α X2 1 3 ξ αh 3 1 3 1 3 i β = − (−1)k   = − −   +   −   2 3 3 k 3 3 2 3 k=1 k 1 2 3 ξ 11α = + · 3 18 Thus,

λ3 = 6β2 − 6β1 + β0 ³ξ 11α´ ³ξ 3α´ α = 6 + − 6 + + ξ + α = · 3 18 2 4 6 Then, 1 τ3 = λ3/λ2 = 3           ξ α X4 1 4 ξ αh 4 1 4 1 4 1 4 i β = − (−1)k   = − −   +   −   +   3 4 4 k 4 4 2 3 4 k=1 k 1 2 3 4 ξ 25α = + · 4 48 Therefore,

λ4 = 20β3 − 30β2 + 12β1 − β0 ³ξ 25α´ ³ξ 11α´ ³ξ 3α´³ ´ = 20 + − 30 + + 12 + ξ + α 4 48 3 18 2 4 125 55 α = 5ξ + α − 10ξ − α + 6ξ + 9α − ξ − α = 12 3 12 1 τ = λ /λ = · 4 4 2 6

69 2.9.3 L-moments for Logistic Distribution

In this subsection, we ﬁnd the ﬁrst four L-moments for the logistic distribution. The logistic distribution has the probability density function([15], Page 196):

α−1e−(1−k)y f(x) = , where y = −k−1 log{1 − k(x − ξ)/α}, (1 + e−y)2

and has the quantile function[17]:

Now, Z 1 Z 1 h i r r βr = x(F )F dF = ξ + α ln F − α ln(1 − F ) F dF 0 0 Z Z ξ 1 1 = + α F r ln F dF − α F r ln(1 − F )dF. (2.9.6) r + 1 0 0 Now, we will ﬁnd Z 1 F r ln F dF. 0 Integrating by parts, we get:

Z 1 ¯ Z 1 F r+1 ln F ¯1 1 −1 F r ln F dF = ¯ − F rdF = · (2.9.7) 0 r + 1 0 r + 1 0 (r + 1) Substituting eq.n.(2.9.5) and eq.n.(2.9.7) in eq.n.(2.9.6), we get:   ξ α α Xr+1 1 r + 1 β = − − (−1)k   . r r + 1 (r + 1)2 r + 1 k k=1 k Then,

β0 = ξ − α − α(−1) = ξ.

70 Therefor,

λ1 = β0 = ξ ξ α α −3 ξ α β = − − ( ) = + 1 2 4 2 2 2 2 ³ξ α´ λ = 2β − β = 2 + − ξ = α 2 1 0 2 2 ξ α 11α ξ α β = − + = + . 2 3 9 18 3 2 ³ξ α´ ³ξ α´ λ = 6β − 6β + β = 6 + − 6 + + ξ 3 2 1 0 3 2 2 2 = 2ξ + 3α − 3ξ − 3α + ξ = 0 0 τ = λ /λ = = 0 3 3 2 α ξ α 25α ξ 11α β = − = = + 3 4 16 48 4 24

λ4 = 20β3 − 30β2 + 12β1 − β0 ³ξ 11α´ ³ξ α´ ³ξ α´ = 20 + − 30 + + 12 + − ξ 4 24 3 2 2 2 55 1 = 5ξ + α − 10ξ − 15α + 6ξ + 6α − ξ = α 6 6 1 τ = λ /λ = · 4 4 2 6

2.9.4 L-moments for Generalized Pareto

We are about to ﬁnd - as in the previous sections - the ﬁrst four L-moments for the generalized pareto distribution. The generalized pareto distribution has the probability density function([15], Page 194):

f(x) = α−1e−(1−k)[−k−1 log{1−k(x−ξ)/α}] , and has the quantile function[17]:

71 Now, we will ﬁnd βr for the generalized pareto distribution:

Z Z Z 1 1 ³ ´ 1 ³ ´k r α r α r βr = x(F )F = ξ + F dF − 1 − F F dF 0 0 k k 0 Z 1 ³ α´ α 1 = ξ + − (1 − F )kF rdF. (2.9.8) r + 1 k k 0 Let u = 1 − F =⇒ du = −dF , F = 1 − u. Then,   Z Z Z 1 0 0 Xr r (1 − F )kF rdF = − uk(1 − u)rdu = − uk (−1)j   ujdu 0 1 1 j=0 j     Z Xr r 0 Xr 1 r = − (−1)j   uk+jdu = (−1)j   . (2.9.9) k + j + 1 j=0 j 1 j=0 j Substituting from eq.n.(2.9.9) in e.qn.(2.9.8) we get :   1 ³ α´ α Xr 1 r β = ξ + − (−1)j   r r + 1 k k k + j + 1 j=0 j

Now, α α³ 1 ´ α α β = ξ + − = ξ + − 0 k k k + 1 k k(k + 1) α = ξ + · k + 1 Thus, α λ = β = ξ + · 1 0 k + 1 Furthermore,     1³ α´ αh 1 1 1 1 i β1 = ξ + −   −   2 k k k + 1 0 k + 2 1 1 α αh 1 1 i 1 α(k + 3) = ξ + − − = ξ + · 2 2k k k + 1 k + 21 2 2(k + 1)(k + 2)

72 So,

h1 α(k + 3) i λ = 2β − β = 2 ξ + 2 1 0 2 2(k + 1)(k + 2) α = · (k + 1)(k + 2)

Moreover,   1³ α´ α X2 1 2 β = ξ + − (−1)j   2 3 k k k + j + 1 j=0 j       1 1 α αh 1 2 1 2 1 2 i = ξ + −   −   +   3 3 k k k + 1 0 k + 1 1 k + 3 2 1 1 α αh 1 2 1 i = + − − + 3 3 k k k + 1 k + 2 k + 3 1 1 α 2α = ξ + − 3 3 k k(k + 1)(k + 2)(k + 3) 1 k2 + 6k + 11 ξ + · 3 3(k + 1)(k + 2)(k + 3)

Then,

λ3 = 6β2 − 6β1 + β0 h1 k2 + 6k + 11 i h1 α(k + 3) i α = 6 ξ + − 6 ξ + + ξ + 3 3(k + 1)(k + 2)(k + 3) 2 2(k + 1)(k + 2) k + 1 2k2 + 12k + 22 3α(k + 3) α = 2ξ + − 3ξ − + ξ + (k + 1)(k + 2)(k + 3) (k + 1)(k + 2) k + 1 2k2 + 12k + 22 − 3(k + 3)2 + (k + 2)(k + 3) = α (k + 1)(k + 2)(k + 3) 1 − k = α· (k + 1)(k + 2)(k + 3)

Also,

1 − k . α 1 − k τ = λ /λ = α = · 3 3 2 (k + 1)(k + 2)(k + 3) (k + 1)(k + 2) k + 3

73 Finally,   1³ α´ α X3 1 3 β = ξ + − (−1)j   3 4 k k k + 1 + j j=0 j         1 α αh 1 3 1 3 1 3 1 3 i = ξ + −   −   +   −   4 4k k k + 1 0 k + 2 1 k + 3 2 k + 4 3 1 α αh 1 3 3 1 i = ξ + − − + − 4 4k k k + 1 k + 2 k + 3 k + 4 1 1 α αh 6 i = ξ + − 4 4 k k (k + 1)(k + 2)(k + 3)(k + 4) 1 k3 + 10k2 + 35k + 50 = ξ + α · 4 4(k + 1)(k + 2)(k + 3)(k + 4)

So,

λ4 = 20β3 − 30β2 + 12β1 − β0 h1 k3 + 10k2 + 35k + 50 i h1 k2 + 6k + 11 i = 20 ξ + α − 30 ξ + 4 4(k + 1)(k + 2)(k + 3)(k + 4) 3 3(k + 1)(k + 2)(k + 3) h1 α(k + 3) i h α i + 12 ξ + − ξ + 2 2(k + 1)(k + 2) k + 1 5k3 + 50k2 + 175k + 250 10k2 + 60k + 110 = 5ξ + α − 10ξ − α (k + 1)(k + 2)(k + 3)(k + 4) (k + 1)(k + 2)(k + 3)(k + 4) 6(k + 3)α α + 6ξ + − ξ − (k + 1)(k + 2) k + 1 k2 − 3k + 2 (1 − k)(2 − k) = α = α · (k + 1)(k + 2)(k + 3)(k + 4) (k + 1)(k + 2)(k + 3)(k + 4)

Therefor,

(1 − k)(2 − k) . α (1 − k)(2 − k) τ = λ /λ = α = · 4 4 2 (k + 1)(k + 2)(k + 3)(k + 4) (k + 1)(k + 2) (k + 3)(k + 4)

74 Chapter 3

ESTIMATION OF L-MOMENTS

In this chapter, we introduce estimation for L-moments, probability weighted moments and L-moment ratios. At the end of this chapter, we introduce the estimation of parameters using L-moments for any distribution with ﬁnite means and we ﬁnd estimations for parameters for some distributions as the Uniform distribution, the Exponential Distribu- tion, Generalized Logistic distribution and Generalized Pareto Distribution.

3.1 The rth Sample L-moments

L-moments have been defined for a probability distributions, but in practice must often be estimated from a finite sample. Estimation is based on a sample of size n, arranged in ascending order [13]. A sample of size 2 contains two observations in ascending order x1:2 and x2:2. The difference between the two observations (x1:2 − x2:2) is a measure of the scale of the distribution. A sample of size 3 contains three observations in ascending order x1:3, x2:3 and x3:3. The difference between the two observation (x2:3 − x1:3) and the difference between the two observation (x3:3 − x2:3) can be subtracted from each other to have a measure of the skewness of the distribution. This leads to: (x3:3 − x2:3) − (x2:3 − x1:3) = x3:3 − 2x2:3 + x1:3. A sample of size 4 contains four observations in ascending order x1:4, x2:4, x3:4 and x4:4. A measure for the kurtosis of the distribution given by:

(x4:4 − x1:4) − 3(x3:4 − x2:4). In short: the above linear combinations of the elements of

75 the order sample contain information about the location, scale, skewness and kurtosis of the distribution from which the sample was drawn [25]. A naturale way to generalize the above approach to sample of size n, is to take all possible sub-samples of size 2 and then take the average of the diﬀerences, i.e.; (x1:2 − x2:2)/2:

 −1 1 n XX ` =   (x − x ). 2 2 i:n j:n 2 i>j

Furthermore, the skewness and kurtosis are similarly obtained as:

 −1 1 n XXX ` =   (x − 2x + x ), 3 3 i:n j:n k:n 3 i>j>k  −1 1 n XXXX ` =   (x − 3x + 3x − x ). 4 4 i:n j:n k:n i:n 4 i>j>k>l

Deﬁnition 3.1.1. ([17]) Let x1, x2, ..., xn be the sample and x1:n ≤ x2:n ≤ ... ≤ xn:n the order sample, and deﬁne the rth sample L-moments to be

 −1   n XX X Xr−1 r − 1   −1 k   `r = ... r (−1) xir−k:n, r = 1, 2, ..., n. r 1≤i1

In particular, X −1 `1 = n xi i  −1 1 n XX ` =   (x − x ), 2 2 i:n j:n 2 i>j  −1 1 n XXX ` =   (x − 2x + x ), 3 3 i:n j:n k:n 3 i>j>k  −1 1 n XXXX ` =   (x − 3x + 3x − x ). 4 4 i:n j:n k:n i:n 4 i>j>k>l

76 Sample L-moments have been used to ﬁnd the estimation of parameters using L-moments

−1 P for any distribution with ﬁnite means. The statistic `1 = n i xi is the sample mean. , The sample L-scale, `2, is a scalar multiple of Gini s mean diﬀerence

 −1 n XX G =   (xi:n − xj:n). 2 i>j

Direct estimators In [21] wang provides direct estimators of L-moment which eliminate the need for introducing PWMs. The estimation procedure follows closely the definition of L-moments by covering all possible combinations in more efficient way. For the sample value xi:n there are (i − 1) values ≤ xi:n and (n − 1) values ≥ x(i:n), and for each subsample of size r, the number of values drawn from each of these categories are considered. The first four direct estimators are given by:

 −1 n Xn `1 =   x(i:n) 1 i=1  −1     1 n Xn h i − 1 n − i i ` =     −   x 2 2 (i:n) 2 i=1 1 1  −1         1 n Xn h i − 1 i − 1 n − i n − i i ` =     − 2     +   x 3 3 (i:n) 3 i=1 2 1 1 2  −1       n Xn h i − 1 i − 1 n − i `4 =     − 3     4 i=1 3 2 1       i − 1 n − i n − i i + 3     −   x(i:n). 1 2 3

77 3.2 The Sample Probability Weighted Moments

In this section, we introduce estimation probability weighted moments and its relation with the estimation of L-moments.

Deﬁnition 3.2.1. [4] The Sample probability weighted moments or probability weighted moments estimators (PWMs estimators), computed for data values x1:n, x2:n, ..., xn:n, arranged in increasing order, are given by:

 −1   Xn −1 n − 1 j − 1 br = n     xj:n, r = 0, 1, 2 ··· . (3.2.1) r j=r+1 r

This may alternatively be written as Xn −1 b0 = n xj:n, j=1 Xn (j − 1) b = n−1 x , 1 (n − 1) j:n j=1 Xn (j − 1)(j − 2) b = n−1 x , 2 (n − 1)(n − 2) j:n j=1 and in general [6] :

Xn (j − 1)(j − 2)...(j − r) b = n−1 x · r (n − 1)(n − 2)...(n − r) j:n j=r+1

Remark 3.2.1. [17] The Sample probability weighted moments br is an unbiased estimator of the probability weighted moment βr.

Proof. Since, Xn (j − 1)(j − 2)...(j − r) b = n−1 x r (n − 1)(n − 2)...(n − r) j:n j=r+1 Xn (j − 1)(j − 2)...(j − r) = x , n(n − 1)(n − 2)...(n − r) j:n j=r+1 then, Xn (j − 1)(j − 2)...(j − r) E(b ) = E(X ). (3.2.2) r n(n − 1)(n − 2)...(n − r) j:n j=r+1

78 Forme eqn.(2.1.2), we have Z n! EX = x[F (x)]j−1[1 − F (x)]n−j dF (x). j:n (j − 1)!(n − j)!

Hence, substituting EXj:n in eq.n.(3.2.2), we have Z Xn (j − 1)(j − 2)...(j − r) n! 1 E(b ) = × x[F (x)]j−1[1 − F (x)]n−j dF (x) r n(n − 1)(n − 2)...(n − r) (j − 1)!(n − j)! j=r+1 0 Z 1 h Xn (n − (r + 1))! i = x [F (x)]j−1[1 − F (x)]n−j dF (x) ((j − (r + 1))! (n − j)! 0 j=r+1   Z 1 h Xn n − r − 1 i = x   [F (x)]j−1[1 − F (x)]n−j dF (x) 0 j=r+1 j − r − 1   Z 1 h Xn n − (r + 1) i = x   [F (x)]j−1[1 − F (x)]n−j dF (x). 0 j=r+1 j − (r + 1) Let k = r + 1. So,   Z 1 h Xn i n − k j−1 n−j E(br) = x   [F (x)] [1 − F (x)] dF (x) 0 j=k j − k   Z 1 h Xn n − k i = x   [F (x)](j−k)+k−1[1 − F (x)]n−(j−k)−k dF (x) 0 j−k=0 j − k   Z 1 h Xn n − k i = x   [F (x)]m+k−1[1 − F (x)]n−m−k dF (x), where m = j − k 0 m=0 m   Z 1 h Xn−k n − k i = xF k−1   [F (x)]m[1 − F (x)]n−k−m dF (x) 0 m=0 m Z 1 h in−k = xF k−1 F (x) + 1 − F (x) dF (x) 0 Z 1 = xF k−1dF (x). 0

Since k = r + 1, then from eq.n.(2.2.1) we have Z 1 r E(br) = x(F ) F dF (x) = βr. (3.2.3) 0

That means, br is an unbiased estimator of the probability weighted moment βr.

79 Note 3.2.1. [17] The sample L-moments `r are linear combination of PWMs estimators, br.

To see this, Xr−1 ∗ `r = pr−1,k bk, r = 1, 2, ..., n; [3] (3.2.4) k=0 where,     r r + k ∗ r−k     pr,k = (−1) . k k

∗ The coeﬃcients pr,k are those of the shifted Legendre polynomials.

Remark 3.2.2. [17]The sample L-moments `r is an unbiased estimator of L-moments λr.

Proof. From eqn.(3.2.4), we have Xr−1 ∗ E(`r) = pr−1,k E(bk), r = 1, 2, ..., n k=0 Xr−1 ∗ = pr−1,k βr, from eq.n.(3.2.3) k=0 = λr, from eq.n.(2.2.10)·

That means, `r is an unbiased estimator of L-moments λr.

The ﬁrst four rth sample L-moments follow from PWMs estimator are [3]:

`1 = b0

`2 = 2b1 − b0

`3 = 6b2 − 6b1 + b0, and

`4 = 20b3 − 30b2 + 12b1 − b0.

Sample L-moments may be used similarly to (conventional) sample moments: they summarize the basic properties-location, scale, skewness, kurtosis-of a data set, they estimate the corresponding properties of the probability distribution from which the data were sampled and they may be used to estimate the parameters of the data were sampled and they may be used to estimate the parameters of the underlying distribution[17].

80 3.3 The rth Sample L-moment Ratios

By dividing the higher-order rth sample L-moments by the dispersion measure, we obtain the rth sample L-moment ratios:

Deﬁnition 3.3.1. [16] Deﬁne the rth sample L-moment ratios to be the quantities

tr = `r/`2, r = 3, 4, 5, ..., (3.3.1)

tr is a natural estimator of τr ([15], Page 28). These are dimensionless quantities, independent of the units of measurement of the data; t3 = `3/`2 is a measure of skewness, t4 = `3/`2 is a measure of kurtosis these are respectively the L-skewness and L-kurtosis. They take values between −1 and +1 (exception: some evenorder L-moment ratios computed from very small samples can be less than -1). The L-moments analogue of the coeﬃcient of variation (standard deviation divided by the mean), is the sample L-CV, deﬁned by:

t = `2/`1 (3.3.2)

t is natural estimators of τ. The estimators tr and t are not unbiased ([15], Page 28).

The quantities `1, `2 (or t), t3, and t4 are useful summary statistics of a sample of data. They can be used to identify the distribution from which a sample was drawn. They can also be used to estimate parameters when ﬁtting a distribution to a sample, by equating the sample and population L-moments [19]. As an example we calculate them for six sets of annual maximum windspeed data taken from simiu, Changery and Filliben (1979). The data are tabled in Table 3.1. The sample L-moments can be calculated using eqn.(3.2.1), eq.n.(3.3.2), eq.n.(3.2.4) and the sample L-moment ratios can be calculated using eq.n.(3.3.2). The results are given in Table 3.2 ([15], Page 30).

81 Table 3.1: Annual maximum windspeed data, in miles per hour, for six sites in the eastern United States. Macon, Ga., 1950-1977. 32 32 34 37 37 38 40 40 40 42 42 42 43 44 45 45 46 48 49 50 51 51 51 53 53 58 58 60 Brownsville, Tex., 1943-1977 32 33 34 34 35 36 37 37 38 38 39 39 40 40 41 41 42 42 43 43 43 44 44 46 46 48 48 49 51 53 53 53 56 63 66 Port Arthur. Tex., 1953-1977 39 43 44 44 45 45 45 46 47 49 51 51 51 51 54 55 55 57 57 60 61 63 66 67 81 Montgomery, Ala., 1950-1977. 34 36 36 37 38 40 40 40 40 40 43 43 43 43 46 46 46 46 47 47 48 49 51 51 51 52 60 77 Key West, Fla., 1958-1976. 35 35 36 36 36 38 42 43 43 46 48 48 52 55 58 64 78 86 90 Corpus Chisti, Tex., 1943-1976. 44 44 44 44 45 45 45 45 46 46 46 47 48 48 48 48 48 49 50 50 50 51 52 55 57 58 60 60 66 67 70 71 77 128

82 Table 3.2: L-moments of the annual maximum windspeed data in Table(3.1)

Site n `1 `2 t t3 t4 Macon 28 45.04 4.46 0.0990 0.0406 0.0838 Brownsvill 35 43.63 4.49 0.1030 0.1937 0.1509 Port Arthur 25 53.08 5.25 0.0989 0.2086 0.1414 Montgomery 28 45.36 4.34 0.0958 0.2316 0.2490 Key West 19 51.00 9.29 0.1821 0.3472 0.1245 Corpus Christi 34 54.47 6.70 0.1229 0.5107 0.3150

Table 3.3: Bais of sample L-CV τ

τ3 0.1 0.2 0.3 0.4 0.5 0.0 -0.001 0.000 0.003 0.009 0.020 0.1 -0.001 -0.001 0.001 0.005 0.014 0.2 -0.001 -0.002 -0.001 0.001 0.008 0.3 -0.001 -0.003 -0.005 -0.004 0.000 0.4 -0.002 -0.006 -0.010 -0.012 -0.011 0.5 -0.003 -0.011 -0.018 -0.025 -0.027 Note: Results are for samples of size 20 from a generalized extreme value distribution

with L-CV τ and L-skewness τ3

83 The bias of the sample L-CV, t, is negligible in sample of size 20 or more. For example, Table 3.3 gives the bias of t for samples of size 20 from a generalized extreme value distribution ([15], Page 28).

3.4 Parameter Estimation Using L-moments

A common problem in statistics is the estimation, from a random sample of size n, of a probability distribution whose specification involves a finite number, p, of unknown parameters. Analogously the usual method of moments, the method of L-moments obtains parameter estimates by equating the first p sample L-moments to the corresponding population quantities. Examples of parameter estimators derived using this method are given in Table 3.4 [17] Method of L-moments [7]: Let F (x) be a distribution function associated with a random variable X and let x(F ) : (0, 1) → R be its quantile function. The rth L-moment of X is given by eq.n.(2.1.12), Z 1 ∗ λr = x(F ) Pr−1(F ) dF, r = 1, 2,.... 0 where     r r + m ∗ r−m     pr,m = (−1) , m m and Xr ∗ ∗ m Pr (F ) = pr,m F . m=0 Theorem 2.4.1 gives the following justification for using L-moments: a) µ1 (mean) is finite if and only if λr exists for all r = 1, 2,... ; b) a distribution F (x) with finite mean µ1 is uniquely characterized by λr for all r = 1, 2,.... L-moments can be used to estimate a finite number of parameters θ ∈ Θ that identify a member of a family of distributions. Suppose {F (x, θ): θ ∈ Θ ⊂ RP }, P a natural

n number, is a family of distributions which is known up to θ. A sample {xi}i=1 is available

84 Table 3.4: [17]Parameter estimation via L-moments for some common distributions Distribution Estimators

Exponential (ξ known)α ˆ = l1 ˆ Gumble αˆ = l2/ log 2, ξ = l1 − γαˆ ˆ Logistic αˆ = l2, ξ = l1 1/2 Normal σˆ = π l2, µˆ = l1 ˆ ˆ Generalized Pareto (ξ known) k = l1/l2 − 2, αˆ = (1 + k)l1 ˆ 2 Generalized extreme value z = 2/(3 + t3) − log 2 log 3, k ≈ 7.8590z + 2.9554z ˆ −kˆ ˆ ˆ ˆ ˆ αˆ = l2k/(1 − 2 )Γ(1 + k), ξ = l1 +α ˆ {Γ(1 + k) − 1}/k ˆ ˆ ˆ ˆ ˆ Generalized logistic k = −t3, αˆ = l2/Γ(1 + k)Γ(1 − k), ξ = l1 + (l2 − αˆ)/k p ³ ´ −1 1+t3 Log-normal z = (8/3)Φ 2 , σˆ ≈ 0.999281z − 0.006118z3 + 0.000127z5 , 2 ˆ 2 µˆ = log {l2/erf(σ/2)} − σˆ /2, ξ = l1 − exp(ˆµ +σ ˆ /2) 1 2 Gamma (ξ known) t = l2/l1; if 0 < t < 2 then z = πt andα ˆ ≈ (1 − 0.3080z)/(z − 0.05812z2 + 0.01765z3);

1 if 2 ≤ t < 1 then z = 1 − t and αˆ ≈ (0.7213z − 0.5947z2)/(1 − 2.1817z + 1.2113z2); ˆ β = l1/αˆ γ is Euler,s constant; Φ−1 is the inverse standard normal distribution function.

85 and the objective is to estimate θ. Since, λr, r = 1, 2, 3,... uniquely characterizes F , θ ˆ may be expressed as a function of λr. Hence, if estimators λr = `r are available, we may P R ˆ ˆ ˆ r ∗ 1 m obtain θ(λ1, λ2,...). From eq.n.(2.2.10), λr = m=0 pr−1,m βm where βm = 0 x(u)u du. th Given the sample, we deﬁne xk,n to be the k smallest element of the sample, such that x1,n ≤ x2,n ≤ ... ≤ xn,n. An unbiased estimator of βr is

Xn (j − 1)(j − 2)...(j − r) βˆ = b = n−1 x r r (n − 1)(n − 2)...(n − r) j:n j=r+1 P ˆ r ∗ ˆ and so λr = m=0 pr−1,m βm [7].

Here are some examples for parameter estimation using method of L-moments:

Example 3.4.1. Uniform Distribution: From Table 2.5, we have the ﬁrst and the second L-moments for the Uniform distribution

1 1 are λ1 = 2 (α + β) and λ2 = 6 (−α + β), then we have two equations

2λ1 = α + β, (3.4.1)

6λ6 = −α + β. (3.4.2)

By solving eq.n.(3.4.1) and (3.4.2) we have:

β = λ1 + 3λ2, α = λ1 − 3λ2. ˆ ˆ ˆ ˆ ˆ Hence, β = λ1 + 3λ2 = `1 + 3`2, αˆ = λ1 − 3λ2 = `1 − 3`2.

Example 3.4.2. Exponential Distribution:

From Table 2.5, we have the ﬁrst L-moments for exponential distribution is: λ1 = ξ + α. ˆ If ξ = 0, then λ1 = α and hence αˆ = λ1 = `1.

Example 3.4.3. Generalized Logistic Distribution From Table 2.5, we have the ﬁrst and the second L-moments and the third ratio L- moments for the Generalized Logistic distribution (respectively) are: λ1 = ξ + α{1 −

86 λ3 Γ(1 + k)Γ(1 − k)}/k, λ2 = αΓ(1 + k)(1 − k), τ3 = −k. Then, k = −τ3 = − . λ2 ˆ ˆ λ2 `3 ˆ ˆ ˆ So k = − = − · Since α = λ2Γ(1 + k)(1 − k), then αˆ = λ2Γ(1 + k)(1 − k). Now, λˆ3 `2 λ2 λ2 ξ = λ1−α{1−Γ(1+k)Γ(1−k)}/k. Since, Γ(1+k)(1−k) = α , then ξ = λ1−α{1− α }/k = α λ2 ˆ ˆ ˆ ˆ λ1 − k + k = λ1 + (λ2 − α)/k · Therefor, ξ = λ1 + (λ2 − αˆ)/k = `1 + (`2 − αˆ)/k ·

Example 3.4.4. Generalized Pareto Distribution From Table 2.5, we have the ﬁrst and the second L-moments for the Generalized pareto distribution are: λ1 = ξ +α/(1+k), λ2 = α/(1+k)(2+k). If ξ = 0, then λ1 = α/(1+k). ˆ ˆ ˆ So, α = (1 + k)λ1. Hence, αˆ = (1 + k)λ1 = (1 + k)`1. α 1 Since, λ2 = α/(1 + k)(2 + k) = (1+k) × (2+k) = λ1/(k + 2), then (k + 2)λ2 = λ1. ˆ So, k = λ1 − 2. Therefor, kˆ = λ1 − 2 = `1 − 2. λ2 λˆ2 `2

87 Chapter 4

Estimation of the Generalized Lambda Distribution from Censored Data

The Generalized Lambda Distribution GLD is a four parameter family of distributions, consisting of a wide variety of curve shapes. The expressions for the PWMs and L- moments of it help us to find out the same for any univariate continuous (both complete and censored) distribution. Another advantage of the use of GLD is that the expressions for the PWMs and L-moments both for complete and censored data do not change, with respect to changes in the form of the distribution, except for the values of the parameters. This makes both analysis and decision making much simpler [23]. In this chapter, we deal with the “Estimation of the Generalized Lambda Distribution from Censored Data”. In the first section, we find the PWMs and L-moments for GLD. In the second section, we discus the PWMs and L-moments for Censored Data (type B for Right Censoring and Left Censoring ). In the third section, we find L-moments for Censored Distributions using GLD. In the last section, we discuss the fitting of the distributions to Censored Data using GLD.

88 4.1 The Family of Generalized Lambda Distribution

The Generalized Lambda Distribution, GLD, is a family of distributions that can take on a very wide range of shapes within one distributional form. The GLD has a number of different applications. Its main use has been in fitting distributions to empirical data, and in the computer generation of different distributions[23]. In this section we introduced the definitions of the quantile function and the probability density function of the Generalized Lambda Distribution, GLD.

Deﬁnition 4.1.1. [2] Distributions belonging to the Generalized Lambda Distribution GLD family are speciﬁed in terms of their quantile function given by

uλ3 − (1 − u)λ4 x(u) = λ1 + , (4.1.1) λ2 where 0 ≤ u ≤ 1 and u = P (X ≤ x) = F (x).

λ1, λ2 are respectively the location and scale parameters and λ3, λ4 are the shape parameters which jointly determines skewness and kurtosis [23].

Deﬁnition 4.1.2. [2] The probability density function of the Generalized Lambda Dis- tribution GLD is given by

λ2 f(x) = (λ −1) (λ −1) , λ3u 3 + λ4(1 − u) 4 where λ1, λ2, λ3, λ4 as given above.

4.2 PWMs and L-moments for GLD

In this section, we find -in general- that the probability weighted moments (PWMs) is the quantities Mk,r,s for GLD, and in special case we find that (PWMs) is βr = M1,r,0 for GLD which enabled us to calculate the L-moments for GLD. Recalling Definition(2.2.1), we have the probability weighted moments (PWMs) of a random variable X with a cumulative distribution function F (X) and quantile function x(F )

89 is the quantities Z 1 © k r sª k r s Mk,r,s = E X [F (X)] [1 − F (X)] = X [F (X)] [1 − F (X)] dF, 0 where k, r, s are real numbers.

A particular useful special cases are the probability weighted moments αr = M1,0,r and βr = M1,r,0. For a distribution that has a quantile function x(u), Z 1 r αr = x(u)(1 − u) du, 0

Z 1 r βr = x(u)u du. (4.2.1) 0 Lemma 4.2.1. n! B(n + 1, a) = Qn , where n is nonnegative integer. (4.2.2) j=0(a + j) Proof. From eq.n.(2.3.2) we have: Γ(n + 1)Γ(a) B(n + 1, a) = Γ(a + n + 1) n! Γ(a) = , from eq.n.(1.4.3) Γ(a + n + 1) n! Γ(a) = , from eq.n.(1.4.2) (a + n)(a + n − 1) ··· (a + 1)aΓ(a) n! = (a + 0)(a + 1) ··· (a + (n − 1))(a + n) n! = Qn · j=0(a + j)

In the next proposition, we ﬁnd the quantities Mk,r,s of GLD.

Proposition 4.2.2. [23] The PWM Mk,r,s of a GLD(λ1, λ2, λ3, λ4) family with quantile function x(u) is given by     Xk k Xi i   k−i −i j   Mk,r,s = λ1 λ2 (−1) B(λ3(i − j) + r + 1, λ4j + s + 1), i=0 i j=0 j

Γ(p)Γ(q) where B(p, q) = Γ(p+q) , (Beta function).

90 Proof.

K r s Mk,r,s = E{X [F (x)] [1 − F (x)] } Z 1 = [x(u)]kur[1 − u]s du Z0 1 h uλ3 − (1 − u)λ4 ik = λ + ur[1 − u]s du 1 λ 0   2 Z k 1 h X k ³uλ3 − (1 − u)λ4 ´i i =   λk−i ur[1 − u]s du λ 1 0 i=0 i 2   k Z X k 1 ³ ´i   k−i −i λ3 λ4 r s = λ1 λ2 u − (1 − u) u [1 − u] du i=0 i 0     k Z i X k 1 h X i ³ ´j i   k−i −i   λ4 λ3 i−j r s = λ1 λ2 − (1 − u) (u ) u [1 − u] du i=0 i 0 j=0 j     Z Xk k Xi i 1   k−i −i j   λ3 i−j λ4 j r s = λ1 λ2 (−1) (u ) ((1 − u) ) u [1 − u] du i=0 i j=0 j 0     Z Xk k Xi i 1   k−i −i j   λ3(i−j)+r λ4j+s = λ1 λ2 (−1) u (1 − u) du, from eq.n.(2.3.2) i=0 i j=0 j 0     Xk k Xi i   k−i −i j   = λ1 λ2 (−1) B(λ3(i − j) + r + 1, λ4j + s + 1)· i=0 i j=0 j

M0,r,0,M0,0,s and M0,r,s do not involve any parameters of the distribution and hence are k of no practical use. From Deﬁnition (2.2.1), the quantities Mk,0,0 = E(X ), (k = 1, 2, ...) are the usual noncentral moments of X. For GLD it is given as     Xk k Xi i   k−i −i j   Mk,0,0 = λ1 λ2 (−1) B(λ3(i − j) + 1, λ4j + 1). i=0 i j=0 j     Xk k Xi i   k−i −i j   Similarly, Mk,r,0 = λ1 λ2 (−1) B(λ3(i − j) + r + 1, λ4j + 1). i=0 i j=0 j     Xk k Xi i   k−i −i j   Mk,0,s = λ1 λ2 (−1) B(λ3(i − j) + 1, λ4j + s + 1). i=0 i j=0 j

91 It is to be noted that     X1 1 Xi i   1−i −i j   M1,0,r = λ1 λ2 (−1) B(λ3(i − j) + 1, λ4j + s + 1) i=0 i j=0 j   1 Xi i = λ B(1, r + 1) + (−1)j   B(λ (i − j) + 1, λ j + s + 1) 1 λ 3 4 2 j=0 j 1 £ ¤ = λ B(1, r + 1) + B(λ + 1, r + 1) − B(1, λ + r + 1) (by Lemma (4.2.1)) 1 λ 3 4 h 2 i λ1 1 1 = + B(λ3 + 1, r + 1) − · r + 1 λ2 λ4 + r + 1

    X1 1 X1 i   1−i −i j   M1,r,0 = λ1 λ2 (−1) B(λ3(i − j) + r + 1, λ4j + 1) i=0 i j=0 j   1 X1 i = λ B(r + 1, 1) + (−1)j   B(λ (i − j) + r + 1, λ j + 1) 1 λ 3 4 2 j=0 j 1 £ ¤ = λ B(r + 1, 1) + B(λ + r + 1, 1) − B(r + 1, λ + 1) 1 λ 3 4 h 2 i λ1 1 1 = + − B(r + 1, λ4 + 1) · r + 1 λ2 λ3 + r + 1

Hence

h i λ1 1 1 M1,r,0 = + − B(r + 1, λ4 + 1) (4.2.3) r + 1 λ2 λ3 + r + 1 h i λ1 1 1 r! = + − Qr (by Lemma (4.2.1))· r + 1 λ2 λ3 + r + 1 j=0(λ4 + j + 1) th Here we consider M1,r,0 only and denote it as βr. From eq.n.(4.2.3) we get the r

PWM βr of a GLD(λ1, λ2, λ3, λ4) family as h i λ1 1 1 βr = + − B(r + 1, λ4 + 1) · (4.2.4) r + 1 λ2 λ3 + r + 1

Now, we are going to ﬁnd β0, β1, β2, β3 of GLD so that we can get the ﬁrst four L- moments for the GLD.

92 Putting r = 0, 1, 2, 3 in eq.n.(4.2.4), we get

1 h 1 i β = λ + − B(1, λ + 1) (4.2.5) 0 1 λ λ + 1 4 2 h 3 i λ1 1 1 β1 = + − B(2, λ4 + 1) (4.2.6) 2 λ2 λ3 + 2 λ 1 h 1 i β = 1 + − B(3, λ + 1) (4.2.7) 2 3 λ λ + 3 4 2 h 3 i λ1 1 1 β3 = + − B(4, λ4 + 1) · (4.2.8) 4 λ2 λ3 + 4 Recalling Deﬁnition 2.1.1, we have the L-moments of X to be the quantities   Xr−1 −1 k r − 1 Lr ≡ r (−1)   EXr−k:r r = 1, 2, .... k=0 k

(Here we used the symbol L instead of λ to distinguish between λ’s of the GLD and that of the L-moments). We can write L-moments from eq.n.(2.2.2) as :     Xr r r + m ∗ ∗ r−m     Lr+1 = pr,mβm, where pr,m = (−1) . (4.2.9) m=0 m m

For example the ﬁrst four L-moments are related to the PWMs as:

L1 = β0,

L2 = 2β1 − β0,

L3 = 6β2 − 6β1 + β0,

L4 = 30β3 − 30β2 + 12β1 − β0.

By giving appropriate values of L1,L2,L3, and L4 corresponding to various distributions, in eq.n.(4.2.4), we can approximate the values of their PWMs and hence L-moments from eq.n. (4.2.9). The expressions for L1,L2,L3, and L4 of some distributions are given in [17] and the numerical values of them obtained by direct calculation are compared with the values obtained from GLD and are given in Table 4.1. Uniform(0,1), Expo- nential(3), Normal(0,1), Pareto(1,5), Logistic(0,1) and Gumbel(0,1) are approximated

93 Table 4.1: [23] Comparison of L-moments Distri x(u) L-moments bution Theoretical Numerical value Direct using GLD

1 2 (α + β) 0.5 0.5 1 Uniform x = α+ 6 (β − α) 0.1667 0.1667 (α, β) (β − α)u 0 0 0 0 0 0 α 3 2.9993 Exp(α) x = −α log(1 − u) α/2 1.5 1.5013 1/3 0.3333 0.3313 1/6 0.1667 0.1670 µ 0 0 Normal x = µ + σφ−1(u) π−1/2σ 0.5642 0.5638 (µ, σ) 0 0 0 √ 30π−1 tan−1 2 − 9 0.1226 0.1245 α/(1 + k) 0.25 0.25 α[1−(1−u)k] Pareto x = k α/(1 + k)(2 + k) 0.1389 0.1389 (α, k) (1 − k)/(3 + k) 0.4286 0.4286 (1 − k)(2 − k)/(3 + k)(4 + k) 0.2481 0.2481 ξ 0 0 Logistic x = ξ+ α 1 0.9989 ¡ ¢ u (ξ, α) α log 1−u 0 0 0 1/6 0.1667 0.1668 ξ + γα 0.5772 0.5775 Gumbel x = ξ− α log 2 0.6931 0.6905 (ξ, α) α log(− log u) 0.1699 0.1699 0.1742 0.1504 0.1504 0.16 (γ is Euler’s constant)

94 respectively by GLD[0.5, 2, 1, 1], GLD[0.02100, −0 : 0003603; −0.4072 ∗ 10−5, −0.001076], GLD[0, 0.1975, 0.1349, 0.1349], GLD[0, −1, 7.34512 ∗ 10−12, −0.2], GLD[0, −0.0003637, −0.0003630, −0.0003637] and GLD[−0.1857, 0.02107, 0.006696, 0.02326]. In column 4 the values in 1st, 2nd, 3rd and 4th rows against each distribution give the numerical values of L1,L2,L3, and L4 respectively of that distribution. The tabled values clearly justify the use of GLD for computing the PWMs and L-moments of unimodal continuous distributions [23].

4.3 PWMs and L-moments for Type I and II Singly Censored Data

This section consists of two subsections. In the first subsection, we introduce the definition of type B right censoring, and then we find the type B PPWMs - which will be defined later in this section - of a right-censoring for GLD. In the other hand, we give the definition of the type B0 left censoring, and then we find the type B0 PPWMs of a left-censoring for GLD. This will be used later in section 4.5 to estimate the parameters of GLD.

Deﬁnition 4.3.1. [23] Observed data sets containing values above or below the analytical threshold of measuring equipment are referred to as censored data.

Such data are frequently encountered in quality and quantity monitoring applications of water, soil, and air samples [28]. In right censoring, the censored observations are greater than the measurement threshold

4.3.1 Case I-Right Censoring

The order statistics of a complete sample of n observations are denoted by the following:

X1:n ≤ X2:n ≤ ... ≤ Xn:n. Type B right censoring occurs when m of these values are observed (m ≤ n) and the

95 remaining n − m are censored above a known threshold T :

X ≤ X ≤ ... ≤ X ≤ T ≤ X ≤ X ≤ ... ≤ X . | 1:n 2:n{z m:n} | m+1:n {zm+2:n n:n} m observed n−m censored The censoring threshold T is the random variable in type B censoring and m is ﬁxed [28].

Example 4.3.1. During the T hours of test we observe r failures (where r can be any number from 0 to n). The (exact) failure times are t1, t2, ..., tr and there are (n − r) units that survived the entire T -hour test without failing. Note that T is ﬁxed in advance and r is random, since we don’t know how many failures will occur until the test is run. Note also that we assume the exact times of failure are recorded when there are failures.

B Type B PPWM, br , is equal to the PWM of the “completed sample,” where the censored observations above the censoring threshold T are set equal to the censoring threshold. It is a well established fact that replacing the censored observations with a ﬁxed value such as the measurement threshold leads to a signiﬁcant bias in the resulting statistics such as the mean, the median, or a quantile [28].

Deﬁnition 4.3.2. [23] The type B PPWM of a right-censored distribution is the ordinary PWM of a (complete) distribution with quantile function   x(u), 0 < u < c; yB(u) = (4.3.1)  x(c), c ≤ u < 1. where T = x(c) is the censoring threshold satisfying P (X ≤ T ) = c.

Note 4.3.1. [20] The type B PPWMs, of a right-censored distribution is given by the relation: Z c r+1 B r 1 − c βr = u x(u)du + x(c)· (4.3.2) 0 r + 1 Proof. The type B PPWMs is obtained from substitution of eq.n.(4.3.1) into eq.n.(4.2.1) leading to

Z 1 B r βr = x(u)u du 0

96 Z 1 = uryB(u) du 0 Z c Z 1 = uryB(u) du + uryB(u) du 0 c Z c Z 1 = urx(u) du + urx(c) du 0 c

Z c Z 1 = urx(u) du + x(c) ur du Z0 c c h ur+1 i1 = urx(u) du + x(c) r + 1 c Z0 c 1 − cr+1 = urx(u) du + x(c). 0 r + 1

Deﬁnition 4.3.3. [23] Deﬁne the incomplete beta function βc(m, n) as: Z m−1 n−1 βc(m, n) = u (1 − u) du, for 0 < c < 1; u > 0. u≤c

Note 4.3.2. We can write the incomplete beta function βc(m, n) as:

Z c m−1 n−1 βc(m, n) = u (1 − u) du. (4.3.3) 0

Note 4.3.3. The incomplete beta function βc(m, n) = B(m, n) Ic(m, n).

Proof. From eq.n.(2.3.4) we have:

Z c 1 m−1 n−1 Ic(m, n) = u (1 − u) du B(m, n) 0 1 = β (m, n), from eq.n.(4.3.3). B(m, n) c

Hence, βc(m, n) = B(m, n) Ic(m, n).

Lemma 4.3.4.

r! Xr r! c(r−j) (1 − c)(a+j) β (r + 1, a) = Q − Q (4.3.4) c r (a + j) (r − j)! j j=0 j=0 i=0(a + i)

97 Proof.

Z c r a−1 βc(r + 1, a) = u (1 − u) du 0

Z 1 Z 1 = ur(1 − u)a−1du − ur(1 − u)a−1du 0 c

Z 1 = B(r + 1, a) − ur(1 − u)a−1du c

Z 1 r! r a−1 = Qr − u (1 − u) du, from eq.n.(4.2.2) (4.3.5) j=0(a + j) c

R 1 r a−1 Now, we want to ﬁnd c u (1 − u) du, using integrating by parts: Let z = ur, dv = (1 − u)a−1du.

r−1 (1−u)a Then, dz = ru du, v = − a . So,

Z 1 ¯ Z 1 h i ur(1 − u)a ¯u=1 (1 − u)a ur(1 − u)a−1du = − ¯ − rur−1 − du. c a u=c c a Hence, Z Z 1 cr(1 − c)a r 1 ur(1 − u)a−1du = + ur−1(1 − u)adu. (4.3.6) c a a c By formula (4.3.6) we have: Z Z 1 cr−1(1 − c)a+1 r − 1 1 ur−1(1 − u)adu = + ur−2(1 − u)a+1du. (4.3.7) c a + 1 a + 1 c

98 Then, substituting eq.n.(4.3.7) int eq.n.(4.3.6) we get: Z Z 1 cr(1 − c)a r hcr−1(1 − c)a+1 r − 1 1 i ur(1 − u)a−1du = + + ur−2(1 − u)a+1du c a a a + 1 a + 1 c Z cr(1 − c)a r[cr−1(1 − c)a+1] r(r − 1) 1 = + + ur−2(1 − u)a+1du a a(a + 1) a(a + 1) c

cr(1 − c)a r[cr−1(1 − c)a+1] r(r − 1)[cr−2(1 − c)a+2] = + + a a(a + 1) a(a + 1)(a + 2)

Z r(r − 1) ... (r − (r − 1)) 1 + ...... + ur−r(1 − u)a+r−1du a(a + 1) ... (a + r − 1) c

cr(1 − c)a r[cr−1(1 − c)a+1] r(r − 1)[cr−2(1 − c)a+2] = + + a a(a + 1) a(a + 1)(a + 2)

h ¯ i r(r − 1) ... (r − (r − 1)) (1 − u)a+r ¯u=1 + ...... + × − ¯ a(a + 1) ... (a + r − 1) a + r u=c

cr(1 − c)a r[cr−1(1 − c)a+1] r(r − 1)[cr−2(1 − c)a+2] = + + a a(a + 1) a(a + 1)(a + 2)

r(r − 1) ... 3 × 2 × 1(1 − c)a+r + ...... + a(a + 1) ... (a + r)

Xr r! c(r−j)(1 − c)(a+j) = Q · (4.3.8) (r − j)! j j=0 i=0(a + i) Substitute (4.3.8) into (4.3.5) we have:

r! Xr r! c(r−j)(1 − c)(a+j) β (r + 1, a) = Q − Q · c r (a + j) (r − j)! j j=0 j=0 i=0(a + i)

In the following proposition, we ﬁnd the type B P P W Ms of a GLD which we use in determining the L-moments of a right-censored distribution.

99 Proposition 4.3.5. [23] The type B P P W Ms of a GLD(λ1, λ2, λ3, λ4) family for singly right censoring are given by

r+1 λ3+r+1 r+1 B λ1c c 1 c βr = + − βc(r + 1, λ4 + 1) + x(c), (4.3.9) r + 1 λ2(λ3 + r + 1) λ2 r + 1

Proof. Z c 1 − cr+1 βB = urx(u)du + x(c) r r + 1 Z0 c h uλ3 − (1 − u)λ4 i 1 − cr+1 = ur λ + du + x(c) 1 λ r + 1 0 Z Z 2 Z c 1 c 1 c 1 − cr+1 r λ3+r λ4 r = λ1 u du + u du − (1 − u) u du + x(c) 0 λ2 0 λ2 0 r + 1 ¯ ¯ ur+1 ¯c 1 uλ3+r+1 ¯c 1 1 − cr+1 = λ1 ¯ + × ¯ − βc(r + 1, λ4 + 1) + x(c) r + 1 0 λ2 λ3 + r + 1 0 λ2 r + 1 r λ cr+1 cλ3+r+1 1 n X r! c(r−j)(1 − c)(λ4+j+1) = 1 + + Q r + 1 λ (λ + r + 1) λ (r − j)! j 2 3 2 j=0 i=0(λ4 + i + 1) r! o 1 − cr+1 − Qr + x(c), from eq.n.(4.3.4). j=0(λ4 + j + 1) r + 1

Now, we are about to find the first four L-moments of a right-censored distribution of the GLD. Before doing so, we have to determine the first four B P P W Ms of the GLD. Putting r = 0, 1, 2, 3 in the expression (4.3.9) and use eq.n.(4.3.4) we get:

λ3+1 B c 1 β0 = λ1c + − βc(1, λ4 + 1) + (1 − c)x(c) λ2(λ3 + 1) λ2 cλ3+1 (1 − c)λ4+1 = λ1c + + + (1 − c)x(c) λ2(λ3 + 1) λ2(λ4 + 1) 2 λ3+2 2 B λ1c c 1 1 − c β1 = + − βc(2, λ4 + 1) + x(c) 2 λ2(λ3 + 2) λ2 2 λ c2 cλ3+2 1 nc(1 − c)λ4+1 (1 − c)λ4+2 − 1 o = 1 + − + 2 λ2(λ3 + 2) λ2 λ4 + 1 (λ4 + 1)(λ4 + 2) 1 − c2 + x(c) 2

100 3 λ3+3 3 B λ1c c 1 1 − c β2 = + − βc(3, λ4 + 1) + x(c) 3 λ2(λ3 + 3) λ2 3 λ c3 cλ3+3 = 1 + 3 λ2(λ3 + 3) 1 nc2(1 − c)λ4+1 2c(1 − c)λ4+2 2(1 − c)λ4+3 − 2 o + + + λ2 λ4 + 1 (λ4 + 1)(λ4 + 2) (λ4 + 1)(λ4 + 2)(λ4 + 3) 1 − c3 + x(c) 3

4 λ3+4 4 B λ1c c 1 1 − c β3 = + − βc(4, λ4 + 1) + x(c) 4 λ2(λ3 + 4) λ2 4 λ c4 cλ3+4 = 1 + 4 λ2(λ3 + 4) 1 nc3(1 − c)λ4+1 3c2(1 − c)λ4+2 6c(1 − c)λ4+3 + + + λ2 λ4 + 1 (λ4 + 1)(λ4 + 2) (λ4 + 1)(λ4 + 2)(λ4 + 3) 6((1 − c)λ4+4 − 1) o + (λ4 + 1)(λ4 + 2)(λ4 + 3)(λ4 + 4) 1 − c4 + x(c)· 4

We can write the L-moments of a right-censored distribution from eq.n.(2.2.2) as follows:     Xr r r + m B ∗ B ∗ r−m     Lr+1 = pr,mβm, where pr,m = (−1) . (4.3.10) m=0 m m

In particular, the ﬁrst four L-moments of the right-censored distribution of the GLD that are related to the type B PPWM of the GLD are:

B B L1 = β0

B B B L2 = 2β1 − β0

B B B B L3 = 6β2 − 6β1 + β0

B B B B B L4 = 30β3 − 30β2 + 12β1 − β0 .

101 4.3.2 Case 2 - Left Censoring

The B0 left censoring results when the observations below a random variable threshold T are censored:

X ≤ X ≤ ... ≤ X ≤ T ≤ X ≤ X ≤ ... ≤ X , | 1:n 2:n {z m−1:n} | m:n m{z+2:n m:n} n−k censored k observed where the number of the censored values (m − 1 = n − k) is ﬁxed [28]. For left censoring, type B0 PPWMs may be derived by replacing the censored observations with the ﬁxed threshold x(c), below which measurements are unavailable [23].

Example 4.3.2. In the ﬁeld of hydrology, left censored data sets arise because river discharges below some measurement threshold are often reported as zero. Such river discharges may have actually been zero or they may have been between zero and the measurement threshold, yet reported as zero [8]. Sometimes it is actually advantageous to in- tentionally censor (or eliminate) observations in order to better understand the frequency and magnitude of ﬂood and drought events [29].

Deﬁnition 4.3.4. [23] The type B0 PPWM of a left-censored distribution is the ordinary PWM of a (complete) distribution with quantile function   x(c), 0 < u < c; yB0 (u) = (4.3.11)  x(u), c ≤ u < 1. where T = x(c) is the censoring threshold satisfying P (X ≤ T ) = c.

Note 4.3.6. [28] The type B0 PPWMs, of a left-censored distribution is given by the relation: r+1 Z 1 B0 c r βr = x(c) + u x(u) du. (4.3.12) r + 1 c Proof. The type B0 PPWMs is obtained from substitution of eq.n.(4.3.11) into eq.n.(4.2.1) leading to

Z 1 B0 r βr = x(u)u du 0

102 Z 1 = uryB0 (u) du 0 Z c Z 1 = uryB0 (u) du + uryB0 (u) du 0 c Z c Z 1 = urx(c) du + urx(u) du 0 c Z c Z 1 = x(c) ur du + urx(u) du 0 Z c h ur+1 ic 1 = x(c) + urx(u) du r + 1 0 Z c 1 cr+1 = urx(u)du + x(c)· c r + 1

In the next proposition, we ﬁnd the type B0 P P W Ms of a GLD by which we determining the L-moments of a left-censored distribution.

0 Proposition 4.3.7. [23] The type B P P W Ms of a GLD(λ1, λ2, λ3, λ4) family for singly left censoring are given by

r+1 λ3+r+1 r+1 B0 λ1(1 − c ) (1 − c ) 1 c βr = + − {B(r+1, λ4 +1)−βc(r+1, λ4 +1)}+ x(c). r + 1 λ2(λ3 + r + 1) λ2 r + 1 (4.3.13) r λ (1 − cr+1) (1 − cλ3+r+1) 1 n X r! c(r−j)(1 − c)(λ4+j+1) o cr+1 = 1 + − Q + x(c). r + 1 λ (λ + r + 1) λ (r − j)! j r + 1 2 3 2 j=0 i=0(λ4 + i + 1) Proof.

Z 1 r+1 B0 r c βr = u x(u)du + x(c) c r + 1 Z 1 h λ3 λ4 i r+1 r u − (1 − u) c = u λ1 + du + x(c) c λ2 r + 1 Z Z Z 1 1 1 1 1 cr+1 r λ3+r λ4 r = λ1 u du + u du − (1 − u) u du + x(c) c λ2 c λ2 c r + 1

¯ ¯ Z 1 Z c ur+1 1 1 uλ3+r+1 1 1 n o cr+1 ¯ ¯ λ4 r λ4 r = λ1 ¯ + × ¯ − (1 − u) u du − (1 − u) u du + x(c) r + 1 c λ2 λ3 + r + 1 c λ2 0 0 r + 1

103 r+1 λ3+r+1 n o r+1 λ1(1 − c ) (1 − c ) 1 c = + − B(r + 1, λ4 + 1) − βc(r + 1, λ4 + 1) + x(c) r + 1 λ2(λ3 + r + 1) λ2 r + 1

r λ (1 − cr+1) (1 − cλ3+r+1) 1 n X r! c(r−j)(1 − c)(λ4+j+1) o = 1 + − Q r + 1 λ (λ + r + 1) λ (r − j)! j 2 3 2 j=0 i=0(λ4 + i + 1)

cr+1 + x(c), from eq.n.(4.3.4). r + 1

Now, we will determine the ﬁrst four B0 P P W Ms of the GLD to determine the ﬁrst four L-moments of a left-censored distribution of the GLD. Putting r = 0, 1, 2, 3 in the expression (4.3.13) we get

λ3+1 B0 (1 − c ) 1 β0 = λ1(1 − c) + − {B(r + 1, λ4 + 1) − βc(1, λ4 + 1)}c.x(c) λ2(λ3 + 1) λ2

2 λ3+2 2 B0 λ1(1 − c ) (1 − c ) 1 c β1 = + − {B(2, λ4 + 1) − βc(2, λ4 + 1)} + x(c) 2 λ2(λ3 + 2) λ2 2

3 λ3+3 3 B0 λ1(1 − c ) (1 − c ) 1 c β2 = + − {B(3, λ4 + 1) − βc(3, λ4 + 1)} + x(c) 3 λ2(λ3 + 3) λ2 3

4 λ3+4 4 B0 λ1(1 − c ) (1 − c ) 1 c β3 = + − {B(4, λ4 + 1) − βc(4, λ4 + 1)} + x(c) 4 λ2(λ3 + 4) λ2 4 We can write L-moments of a left-censored distribution from eqn.(2.2.2) as :     Xr r r + m B0 ∗ B0 ∗ r−m     Lr+1 = pr,mβm , where pr,m = (−1) . (4.3.14) m=0 m m In particular, the ﬁrst four L-moments of the left-censored distribution of the GLD that are related to the type B0 PPWM of the GLD are:

B0 B0 L1 = β0

B0 B0 B0 L2 = 2β1 − β0

B0 B0 B0 B0 L3 = 6β2 − 6β1 + β0

B0 B0 B0 B0 B0 L4 = 30β3 − 30β2 + 12β1 − β0 .

104 4.4 L-moments for Censored Distributions Using GLD

In section 4.3, we ﬁnd the type B PPWMs for GLD. This will be used to ﬁnd the B PPWMs for Pareto distribution, which is considered to be a special case of GLD and its quantile function will be given soon. The L-moments for Pareto distribution is deduced later.

Deﬁnition 4.4.1. [23] The Pareto distribution has the quantile function:

x(u) = α[1 − (1 − u)k]/k. h i 1−(1−c)r+k Assume that mr = α r+k . We will use this assumption to express the L- moments of Pareto distribution. Now, we are going to calculate the ﬁrst four L-moments for Pareto distribution.

B First, we calculate β0 for Pareto distribution:

λ3+1 λ4+1 B c (1 − c) β0 = λ1c + + + (1 − c)x(c) λ2(λ3 + 1) λ2(λ4 + 1)

c (1 − c)k+1 α 1 α = + + − + (1 − c)[1 − (1 − c)k] k k k(k + 1) k k α α α (k + 1)

αh (1 − c)k+1 1 i = c + − + (1 − c) − (1 − c)k+1 k k + 1 k + 1

αh(1 − c)k+1 − 1 i = + 1 − (1 − c)k+1 k k + 1

αh i³ −1 ´ = 1 − (1 − c)k+1 + 1 k k + 1

αh i³−1 + k + 1´ = 1 − (1 − c)k+1 k k + 1 αh i³ k ´ = 1 − (1 − c)k+1 k k + 1

h1 − (1 − c)k+1 i = α = αm . k + 1 1

105 Thus, the 1st L-moment for Pareto distribution is given by:

1 B LB = β0 = αm1.

B Second, we calculate β1 for Pareto distribution:

2 λ3+2 n λ4+1 λ4+2 o B λ1c c 1 c(1 − c) (1 − c) − 1 β1 = + − + 2 λ2(λ3 + 2) λ2 λ4 + 1 (λ4 + 1)(λ4 + 2)

1 − c2 + x(c) 2

c2 1 nc(1 − c)k+1 (1 − c)k+2 − 1o 1 − c2 h iα = + + + 1 − (1 − c)k k k/α k + 1 (k + 1)(k + 2) 2 k 2( α )

αhc2 c(1 − c)k+1 (1 − c)k+2 − 1 1 − c2 1 − c2 i = + + + − (1 − c)k k 2 k + 1 (k + 1)(k + 2) 2 2

αhc2 c(1 − c)k+1 (1 − c)k+2 − 1 1 c2 1 + c i = + + + − − (1 − c)k+1 k 2 k + 1 (k + 1)(k + 2) 2 2 2

αhc(1 − c)k+1 (1 − c)k+2 − 1 1 1 + c i = + + − (1 − c)k+1 k k + 1 (k + 1)(k + 2) 2 2

αhc(1 − c)k+1 (1 − c)k+2 1 1 1 + c i = + − + − (1 − c)k+1 k k + 1 (k + 1)(k + 2) (k + 1)(k + 2) 2 2

αh2(k + 2)c(1 − c)k+1 + 2(1 − c)k+2 − 2 + (k + 1)(k + 2) = k 2(k + 1)(k + 2)

−(k + 1)(k + 2)(1 + c)(1 − c)k+1 i + 2(k + 1)(k + 2)

αh2(k + 2)c(1 − c)k+1 + 2(1 − c)k+2 − 2 + k2 + 3k + 2 = k 2(k + 1)(k + 2)

−(k + 1)(k + 2)(1 + c)(1 − c)k+1 i + 2(k + 1)(k + 2)

106 £ ¤ αh(1 − c)k+1 2(k + 2)c − (k + 1)(k + 2)(1 + c) + 2(1 − c)k+2 + k2 + 3k i = k 2(k + 1)(k + 2) £ ¤ αh(1 − c)k+1 2(k + 2)c − (k + 1)(k + 2) − (k + 1)(k + 2)c + 2(1 − c)k+2 + k2 + 3k i = k 2(k + 1)(k + 2) £ © ª ¤ αh(1 − c)k+1 c 2(k + 2) − (k + 1)(k + 2) − (k + 1)(k + 2) + 2(1 − c)k+2 + k2 + 3k i = k 2(k + 1)(k + 2) £ © ª ¤ αh(1 − c)k+1 c(k + 2) 2 − (k + 1) − (k + 1)(k + 2) + 2(1 − c)k+2 + k2 + 3k i = k 2(k + 1)(k + 2) £ ¤ αh(k + 2)(1 − c)k+1 c(−k + 1) − (k + 1) + 2(1 − c)k+2 + k2 + 3k i = k 2(k + 1)(k + 2) £ ¤ αh(k + 2)(1 − c)k+1 c(−k + 1) − (−k + 1) + (−k + 1) − (k + 1) = k 2(k + 1)(k + 2) 2(1 − c)k+2 + k2 + 3k i + 2(k + 1)(k + 2) £ ¤ αh(k + 2)(1 − c)k+1 (−k + 1)(c − 1) − 2k + 2(1 − c)k+2 + k2 + 3k i = k 2(k + 1)(k + 2)

αh−(k + 2)(−k + 1)(1 − c)k+2 − 2k(k + 2)(1 − c)k+1 + 2(1 − c)k+2 + k2 + 3k i = k 2(k + 1)(k + 2) £ ¤ αh(1 − c)k+2 − (k + 2)(−k + 1) + 2 − 2k(k + 2)(1 − c)k+1 + k2 + 3k i = k 2(k + 1)(k + 2)

αh(1 − c)k+2(k2 + k) − 2k(k + 2)(1 − c)k+1 + k2 + 3k i = . k 2(k + 1)(k + 2)

αh(1 − c)k+2(k2 + k) − (k2 + k) + (k2 + k) = k 2(k + 1)(k + 2)

−2k(k + 2)(1 − c)k+1 + 2k(k + 2) − 2k(k + 2) + k2 + 3k i + 2(k + 1)(k + 2)

107 αh(k2 + k)[(1 − c)k+2 − 1] − 2k(k + 2)[(1 − c)k+1 − 1] = k 2(k + 1)(k + 2)

k2 + k − 2k2 − 4k + k2 + 3k i h 1 − (1 − c)k+2 1 − (1 − c)k+1 i + + α − + . 2(k + 1)(k + 2) 2(k + 2) (k + 1)

Thus, the 2nd L-moment for Pareto distribution is given by:

B B B L2 = 2β1 − β0 h 1 − (1 − c)k+2 2[1 − (1 − c)k+1] 1 − (1 − c)k+1 i = α − + − (k + 2) (k + 1) (k + 1)

h1 − (1 − c)k+1 1 − (1 − c)k+2 i = α − (k + 1) (k + 2)

= α(m1 − m2)·

B Third, we calculate β2 for Pareto distribution:

3 λ3+3 n 2 λ4+1 λ4+2 B λ1c c 1 c (1 − c) 2c(1 − c) β2 = + + + 3 λ2(λ3 + 3) λ2 λ4 + 1 (λ4 + 1)(λ4 + 2)

2(1 − c)λ4+3 − 2 o 1 − c3 + + x(c) (λ4 + 1)(λ4 + 2)(λ4 + 3) 3

c3 1 nc2(1 − c)k+1 2c(1 − c)k+2 2(1 − c)k+3 − 2 o = + + + k k/α k + 1 (k + 1)(k + 2) (k + 1)(k + 2)(k + 3) 3( α )

1 − c3 h iα + 1 − (1 − c)k 3 k

αhc3 c2(1 − c)k+1 2c(1 − c)k+2 2(1 − c)k+3 − 2 = + + + k 3 k + 1 (k + 1)(k + 2) (k + 1)(k + 2)(k + 3)

1 − c3 1 − c3 i + − (1 − c)k 3 3

108 αh1 {(1 − c)2 − 2(1 − c) + 1}(1 − c)k+1 2{−(1 − c) + 1}(1 − c)k+2 = + + k 3 k + 1 (k + 1)(k + 2)

2(1 − c)k+3 − 2 {(1 − c)3 − 3(1 − c)2 + 3(1 − c)} i + − (1 − c)k (k + 1)(k + 2)(k + 3) 3

αh1 (1 − c)k+3 − 2(1 − c)k+2 + (1 − c)k+1 −2(1 − c)k+3 + 2(1 − c)k+2 = + + k 3 k + 1 (k + 1)(k + 2)

2(1 − c)k+3 − 2 −(1 − c)k+3 + 3(1 − c)k+2 − 3(1 − c)k+1 i + + (k + 1)(k + 2)(k + 3) 3

αh (k + 1)(k + 2)(k + 3) 3(k + 2)(k + 3)(1 − c)k+3 − 6(k + 2)(k + 3)(1 − c)k+2 = + k 3(k + 1)(k + 2)(k + 3) 3(k + 1)(k + 2)(k + 3)

3(k + 2)(k + 3)(1 − c)k+1 + 3(k + 1)(k + 2)(k + 3)

−6(k + 3)(1 − c)k+3 + 6(k + 3)(1 − c)k+2 6(1 − c)k+3 − 6 + + 3(k + 1)(k + 2)(k + 3) 3(k + 1)(k + 2)(k + 3)

−(k + 1)(k + 2)(k + 3)(1 − c)k+3 + 3(k + 1)(k + 2)(k + 3)(1 − c)k+2 + 3(k + 1)(k + 2)(k + 3)

−3(k + 1)(k + 2)(k + 3)(1 − c)k+1 i + 3(k + 1)(k + 2)(k + 3) £ ¤ αh(k + 1)(k + 2)(k + 3) − 6 (1 − c)k+1 3(k + 2)(k + 3) − 3(k + 1)(k + 2)(k + 3) = + k 3(k + 1)(k + 2)(k + 3) 3(k + 1)(k + 2)(k + 3) £ ¤ (1 − c)k+2 − 6(k + 2)(k + 3) + 6(k + 3) + 3(k + 1)(k + 2)(k + 3) + 3(k + 1)(k + 2)(k + 3) £ ¤ (1 − c)k+3 3(k + 2)(k + 3) − 6(k + 3) + 6 − (k + 1)(k + 2)(k + 3) i + 3(k + 1)(k + 2)(k + 3)

109 αh k3 + 6k2 + 11k 3(k + 2)(k + 3)(1 − c)k+1(1 − k − 1) = + k 3(k + 1)(k + 2)(k + 3) 3(k + 1)(k + 2)(k + 3) £ ¤ 3(k + 3)(1 − c)k+2 − 2(k + 2) + 2 + (k + 1)(k + 2) + 3(k + 1)(k + 2)(k + 3) £ ¤ (1 − c)k+3 3k2 + 15k + 18 − 6k − 18 + 6 − k3 − 6k2 − 11k − 6 i + 3(k + 1)(k + 2)(k + 3)

αh k3 + 6k2 + 11k −3k(k + 2)(k + 3)(1 − c)k+1 = + k 3(k + 1)(k + 2)(k + 3) 3(k + 1)(k + 2)(k + 3)

3(k + 3)(1 − c)k+2(−2k − 4 + 2 + k2 + 3k + 2) (1 − c)k+3(−k3 − 3k2 − 2k)i + + 3(k + 1)(k + 2)(k + 3) 3(k + 1)(k + 2)(k + 3)

αh k3 + 6k2 + 11k −3k(k + 2)(k + 3)(1 − c)k+1 = + k 3(k + 1)(k + 2)(k + 3) 3(k + 1)(k + 2)(k + 3)

3k(k + 1)(k + 3)(1 − c)k+2 −k(k + 1)(k + 2)(1 − c)k+3 i + + 3(k + 1)(k + 2)(k + 3) 3(k + 1)(k + 2)(k + 3)

αh k3 + 6k2 + 11k = k 3(k + 1)(k + 2)(k + 3)

−3k(k + 2)(k + 3)(1 − c)k+1 + 3k(k + 2)(k + 3) − 3k(k + 2)(k + 3) + 3(k + 1)(k + 2)(k + 3)

3k(k + 1)(k + 3)(1 − c)k+2 − 3k(k + 1)(k + 3) + 3k(k + 1)(k + 3) + 3(k + 1)(k + 2)(k + 3)

−k(k + 1)(k + 2)(1 − c)k+3 + k(k + 1)(k + 2) − k(k + 1)(k + 2)i + 3(k + 1)(k + 2)(k + 3)

αh k3 + 6k2 + 11k = k 3(k + 1)(k + 2)(k + 3)

3k(k + 2)(k + 3)[1 − (1 − c)k+1] − 3k(k + 2)(k + 3) + 3(k + 1)(k + 2)(k + 3)

110 −3k(k + 1)(k + 3)[1 − (1 − c)k+2] + 3k(k + 1)(k + 3) + 3(k + 1)(k + 2)(k + 3)

k(k + 1)(k + 2)[1 − (1 − c)k+3] − k(k + 1)(k + 2)i + 3(k + 1)(k + 2)(k + 3)

αhk3 + 6k2 + 11k − 3k(k + 2)(k + 3) + 3k(k + 1)(k + 3) − k(k + 1)(k + 2) = k 3(k + 1)(k + 2)(k + 3)

k[1 − (1 − c)k+1] −k[1 − (1 − c)k+2] k[1 − (1 − c)k+3]i + + + (k + 1) (k + 2) 3(k + 3)

h1 − (1 − c)k+1 1 − (1 − c)k+2 1 − (1 − c)k+3 i = α − + . (k + 1) (k + 2) 3(k + 3)

Thus, the 3ed L-moment for Pareto distribution is given by:

B B B B L3 = 6β2 − 6β1 + β0

h 1 − (1 − c)k+1 1 − (1 − c)k+2 1 − (1 − c)k+3 1 − (1 − c)k+2 = α 6 − 6 + 2 + 3 (k + 1) (k + 2) (k + 3) (k + 2)

1 − (1 − c)k+1 1 − (1 − c)k+1 i − 6 + (k + 1) (k + 1)

h1 − (1 − c)k+1 1 − (1 − c)k+2 1 − (1 − c)k+3 i = α − 3 + (k + 1) (k + 2) (k + 3)

= α(m1 − 3m2 + 2m3).

111 B Finally, we calculate β3 for Pareto distribution:

4 λ3+4 B λ1c c β3 = + 4 λ2(λ3 + 4)

1 nc3(1 − c)λ4+1 3c2(1 − c)λ4+2 6c(1 − c)λ4+3 + + + λ2 λ4 + 1 (λ4 + 1)(λ4 + 2) (λ4 + 1)(λ4 + 2)(λ4 + 3) 6((1 − c)λ4+4 − 1) o 1 − c4 + + x(c) (λ4 + 1)(λ4 + 2)(λ4 + 3)(λ4 + 4) 4

c4 1 nc3(1 − c)k+1 3c2(1 − c)k+2 6c(1 − c)k+3 = + + + k k/α k + 1 (k + 1)(k + 2) (k + 1)(k + 2)(k + 3) 4( α )

6(1 − c)k+4 − 6 o 1 − c4 h iα + + 1 − (1 − c)k (k + 1)(k + 2)(k + 3)(k + 4) 4 k

αhc4 c3(1 − c)k+1 3c2(1 − c)k+2 6c(1 − c)k+3 = + + + k 4 k + 1 (k + 1)(k + 2) (k + 1)(k + 2)(k + 3)

6(1 − c)k+4 − 6 1 − c4 £ ¤i + + 1 − (1 − c)k (k + 1)(k + 2)(k + 3)(k + 4) 4

αhc4 c3(1 − c)k+1 3c2(1 − c)k+2 6c(1 − c)k+3 = + + + k 4 k + 1 (k + 1)(k + 2) (k + 1)(k + 2)(k + 3)

6(1 − c)k+4 − 6 1 − c4 1 − c4 i + + − (1 − c)k (k + 1)(k + 2)(k + 3)(k + 4) 4 4

αh1 c3(1 − c)k+1 3c2(1 − c)k+2 6c(1 − c)k+3 = + + + k 4 k + 1 (k + 1)(k + 2) (k + 1)(k + 2)(k + 3)

6(1 − c)k+4 − 6 1 − c4 i + − (1 − c)k (k + 1)(k + 2)(k + 3)(k + 4) 4

112 αh1 {−(1 − c)3 + 3(1 − c)2 − 3(1 − c) + 1}(1 − c)k+1 = + k 4 k + 1

3{(1 − c)2 − 2(1 − c) + 1}(1 − c)k+2 + (k + 1)(k + 2)

6{−(1 − c) + 1}(1 − c)k+3 6(1 − c)k+4 − 6 + + (k + 1)(k + 2)(k + 3) (k + 1)(k + 2)(k + 3)(k + 4)

−(1 − c)4 + 4(1 − c)3 − 6(1 − c)2 + 4(1 − c) i − (1 − c)k 4

αh1 −(1 − c)k+4 + 3(1 − c)k+3 − 3(1 − c)k+2 + (1 − c)k+1 = + k 4 k + 1

3(1 − c)k+4 − 6(1 − c)k+3 + 3(1 − c)k+2 + (k + 1)(k + 2)

−6(1 − c)k+4 + 6(1 − c)k+3 6(1 − c)k+4 − 6 + + (k + 1)(k + 2)(k + 3) (k + 1)(k + 2)(k + 3)(k + 4)

(1 − c)k+4 − 4(1 − c)k+3 + 6(1 − c)k+2 − 4(1 − c)k+1 i + 4

αh (k + 1)(k + 2)(k + 3)(k + 4) = k 4(k + 1)(k + 2)(k + 3)(k + 4)

4(k + 2)(k + 3)(k + 4){−(1 − c)k+4 + 3(1 − c)k+3 − 3(1 − c)k+2 + (1 − c)k+1} + 4(k + 1)(k + 2)(k + 3)(k + 4)

4(k + 3)(k + 4){3(1 − c)k+4 − 6(1 − c)k+3 + 3(1 − c)k+2} + 4(k + 1)(k + 2)(k + 3)(k + 4)

−24(k + 4)(1 − c)k+4 + 24(k + 4)(1 − c)k+3 24(1 − c)k+4 − 24} + + 4(k + 1)(k + 2)(k + 3)(k + 4) 4(k + 1)(k + 2)(k + 3)(k + 4)

(k + 1)(k + 2)(k + 3)(k + 4){(1 − c)k+4 − 4(1 − c)k+3 + 6(1 − c)k+2 − 4(1 − c)k+1}i + 4(k + 1)(k + 2)(k + 3)(k + 4)

113 αh(k + 1)(k + 2)(k + 3)(k + 4) − 24 = k 4(k + 1)(k + 2)(k + 3)(k + 4) £ ¤ (1 − c)k+1 4(k + 2)(k + 3)(k + 4) − 4(k + 1)(k + 2)(k + 3)(k + 4) + 4(k + 1)(k + 2)(k + 3)(k + 4) £ (1 − c)k+2 − 12(k + 2)(k + 3)(k + 4) + 12(k + 3)(k + 4) + 4(k + 1)(k + 2)(k + 3)(k + 4) ¤ 6(k + 1)(k + 2)(k + 3)(k + 4) + 4(k + 1)(k + 2)(k + 3)(k + 4) £ (1 − c)k+3 12(k + 2)(k + 3)(k + 4) − 24(k + 3)(k + 4) + 24(k + 4) + 4(k + 1)(k + 2)(k + 3)(k + 4) ¤ −4(k + 1)(k + 2)(k + 3)(k + 4) + 4(k + 1)(k + 2)(k + 3)(k + 4)

£ (1 − c)k+4 − 4(k + 2)(k + 3)(k + 4) + 12(k + 3)(k + 4) − 24(k + 4) + 24 + 4(k + 1)(k + 2)(k + 3)(k + 4) ¤ (k + 1)(k + 2)(k + 3)(k + 4) i + 4(k + 1)(k + 2)(k + 3)(k + 4)

αh(k + 1)(k + 2)(k + 3)(k + 4) − 24 −4k(k + 2)(k + 3)(k + 4)(1 − c)k+1 = + k 4(k + 1)(k + 2)(k + 3)(k + 4) 4(k + 1)(k + 2)(k + 3)(k + 4)

6k(k + 1)(k + 3)(k + 4)(1 − c)k+2 −4k(k + 1)(k + 2)(k + 4)(1 − c)k+3 + + 4(k + 1)(k + 2)(k + 3)(k + 4) 4(k + 1)(k + 2)(k + 3)(k + 4)

k(k + 1)(k + 2)(k + 3)(1 − c)k+4 i + 4(k + 1)(k + 2)(k + 3)(k + 4)

114 αh k4 + 10k3 + 35k2 + 50k = k 4(k + 1)(k + 2)(k + 3)(k + 4) £ ¤ 4k(k + 2)(k + 3)(k + 4) 1 − (1 − c)k+1 − 4k(k + 2)(k + 3)(k + 4) + 4(k + 1)(k + 2)(k + 3)(k + 4) £ ¤ −6k(k + 1)(k + 3)(k + 4) 1 − (1 − c)k+2 + 6k(k + 1)(k + 3)(k + 4) + 4(k + 1)(k + 2)(k + 3)(k + 4) £ ¤ 4k(k + 1)(k + 2)(k + 4) 1 − (1 − c)k+3 − 4k(k + 1)(k + 2)(k + 4) + 4(k + 1)(k + 2)(k + 3)(k + 4) £ ¤ −k(k + 1)(k + 2)(k + 3) 1 − (1 − c)k+4 + k(k + 1)(k + 2)(k + 3)i + 4(k + 1)(k + 2)(k + 3)(k + 4)

hk4 + 10k3 + 35k2 + 50k − 4k(k + 2)(k + 3)(k + 4) + 6k(k + 1)(k + 3)(k + 4) = α 4k(k + 1)(k + 2)(k + 3)(k + 4)k(k + 1)(k + 2)(k + 3)

−4k(k + 1)(k + 2)(k + 4) + k(k + 1)(k + 2)(k + 3) + 4k(k + 1)(k + 2)(k + 3)(k + 4)k(k + 1)(k + 2)(k + 3) £ ¤ 1 − (1 − c)k+1 3 1 − (1 − c)k+2 1 − (1 − c)k+3 1 − (1 − c)k+4 i + − + − (k + 1) 2(k + 2) (k + 3) 4(k + 4) £ ¤ h1 − (1 − c)k+1 3 1 − (1 − c)k+2 1 − (1 − c)k+3 1 − (1 − c)k+4 i = α − + − . (k + 1) 2(k + 2) (k + 3) 4(k + 4) Thus, the 4th L-moment for Pareto distribution is given by:

B L4 = 20β3 − 30β2 + 12β1 − β0

h 1 − (1 − c)k+1 1 − (1 − c)k+2 1 − (1 − c)k+3 1 − (1 − c)k+4 = α 20 − 30 + 20 − 5 (k + 1) (k + 2) (k + 3) 4(k + 4)

1 − (1 − c)k+1 1 − (1 − c)k+2 1 − (1 − c)k+3 1 − (1 − c)k+2 − 30 + 30 − 10 − 6 (k + 1) (k + 2) (k + 3) (k + 2)

1 − (1 − c)k+1 1 − (1 − c)k+1 i + 12 − (k + 1) (k + 1)

115 h1 − (1 − c)k+1 1 − (1 − c)k+2 1 − (1 − c)k+3 1 − (1 − c)k+4 i = α − 6 + 10 − 5 (k + 1) (k + 2) (k + 3) 4(k + 4)

h i = α m1 − 6m2 + 10m3 − 5m4 .

In Table 4.2 the numerical values of the ﬁrst four L-moments of Pareto distribution for diﬀerent censoring values are compared with the values obtained by the corresponding GLD approximation [23].

116 Table 4.2: :[23] L-moment of Pareto distribution for censoring fraction c B B B B Distribution c method L1 L2 L3 L4

0.99 direct 0.2437 0.1326 0.0533 0.0283 gld 0.2437 0.1326 0.0533 0.0283

0.9 direct 0.2104 0.1010 0.0250 0.0043 gld 0.2104 0.1010 0.0250 0.0043

0.8 direct 0.1810 0.0760 0.0074 -0.0050 gld 0.1810 0.0760 0.0074 -0.0050

Pareto 0.7 direct 0.1546 0.0562 -0.0026 -0.0064 α = 1/5, gld 0.1546 0.0562 -0.0026 -0.0064 k = −1/5 0.6 direct 0.1299 0.0401 -0.0075 -0.0043 gld 0.1299 0.0401 -0.0075 -0.0043

0.5 direct 0.1064 0.02772 -0.0089 -0.0013 gld 0.1064 0.0272 -0.0089 -0.0013

Table (4.2) strongly recommends the use of GLD for modeling univariate continuous distributions using their PWMs and L-moments even for censored observations [23].

117 4.5 Fitting of the Distributions to Censored Data Us- ing GLD

In this section, we estimate type B PPWMs for right censoring distribution and type B0 PPWMs for left censoring distribution. Using this estimations, we can estimate the parameters of the right and the left censoring GLD.

PPWM Estimators for Right Censoring Type B PPWM is computed from the completed sample, where the n−m censored values in X ≤ X ≤ ... ≤ X ≤ T ≤ X ≤ X ≤ ... ≤ X ≤ X | 1:n 2:n{z m:n} | m+1:n m+2:n{z n−1:n n:n} m−1 observed n−m censored are replaced by the censoring threshold T [28].

Deﬁnition 4.5.1. [28] Let x ≤ x ≤ ... ≤ x ≤ T ≤ x ≤ x ≤ ... ≤ x ≤ x | 1:n 2:n{z m:n} | m+1:n m+2:n{z n−1:n n:n} m−1 observed n−m censored be an order sample, and deﬁne the sample type B PPWM for right censoring distribution,

B br as:

1 n Xm (j − 1)(j − 2)...(j − r) ³ Xn (j − 1)(j − 2)...(j − r) ´ o bB = x + T . (4.5.1) r n (n − 1)(n − 2)...(n − r) j:n (n − 1)(n − 2)...(n − r) j=1 j=m+1

B Note 4.5.1. [28] The samples type B PPWMs, br are unbiased estimators of the type B B PPWM βr for r = 1, 2, 3,...

To estimate the parameters of the right censored GLD in the case of type one single censoring, we can equate the sample and population PPWMs. As for estimation usually B type PPWMs are preferred by comparing the ﬁrst four theoretical and sample moments obtained from expressions (4.3.2) and (4.5.1), we can obtain the appropriate values of the parameters λ1, λ2, λ3 and λ4 [20].

118 PPWM Estimators for Left Censoring Type B0 PPWM is computed from the completed sample, where the n−k censored values in X ≤ X ≤ ... ≤ X ≤ T ≤ X ≤ X ≤ ... ≤ X , | 1:n 2:n {z m−1:n} | m:n m{z+2:n n:n} (m−1=n−k) n−k censored k observed are replaced by the censoring threshold T [28].

Deﬁnition 4.5.2. [28] Let x ≤ x ≤ ... ≤ x ≤ T ≤ x ≤ x ≤ ... ≤ x , | 1:n 2:n {z m−1:n} | m:n m{z+2:n n:n} n−k censored k observed (m − 1 = n − k) ba an order sample and deﬁne the sample type B0 PPWM for lift cen-

B0 soring distribution, br as:

n n−k n o 0 1 X (j − 1)(j − 2)...(j − r) X (j − 1)(j − 2)...(j − r) bB = T + x . (4.5.2) r n (n − 1)(n − 2)...(n − r) (n − 1)(n − 2)...(n − r) j:n j=1 j=n−k+1

B0 0 B0 Note 4.5.2. [23] The sampls type br are unbiased estimators of the type B , βr for r = 1, 2, 3,...

In eq.n.(4.5.2), k = n − m + 1. In the case of type B0 censoring T is to be replaced by Xm:n in the above expressions. So, by comparing the First 4 theoretical and sample PPWMs using expressions (4.3.13) and (4.5.2), we can ﬁt a GLD for a left censored data [23].

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