Available online at http://www.ajol.info/index.php/njbas/index ISSN 0794-5698 Nigerian Journal of Basic and Applied Science (December, 2017), 25(2): 130-142 DOI: http://dx.doi.org/10.4314/njbas.v25i2.14
The Gamma-Rayleigh Distribution and Applications to Survival Data
*1E. E. E. Akarawak, 2I. A. Adeleke, 3R. O. Okafor 1,3Department of Mathematics, University of Lagos, Akoka (Nigeria)
2Department of Actuarial Science and Insurance, University of Lagos, Akoka (Nigeria) Full Length Research Article Research Length Full [*Corresponding Author: Email: [email protected]; : +2348137678288] ABSTRACT Studies on probability distribution functions and their properties are needful as they are very important in modeling random phenomena. However, research has shown that some real life data can be modeled more adequately by distributions obtained as combination of two random variables with known probability distributions. This paper introduces the Gamma-Rayleigh distribution (GRD) as a new member of the Gamma-X family of generalized distributions. The Transformed-Transformer method is used to combine the Gamma and Rayleigh distributions. Various properties of the resulting two- parameter Gamma-Rayleigh distribution, including moments, moment generating function, survival function and hazard function are derived. Results of simulation study reveals that the distribution is unimodal, skewed and normal-type for some values of the shape parameter. The distribution is also found to relate with the Gamma, Rayleigh and Generalized-Gamma distributions. The method of maximum likelihood has been used to estimate the shape and scale parameters of the distribution. To illustrate its adequacy in modelling real life data the distribution is fitted to two survival data sets. The results show that the distribution produced fits that are competitive and compared better, in some cases, to the Gamma, Rayleigh, Weibull and Lognormal distributions. Keywords: Gamma-X family, Gamma-Rayleigh distribution, Maximum Likelihood estimators, Survival data.
INTRODUCTION Gupta and Kundu, 2001; Pal et al., 2006) have Studies on probability distribution functions and shown that distributions of combined random their properties are needful as they are very variables are more flexible, perform better and important in modeling random phenomena. have wider applicability. However, research has shown that some real life data that cannot be modeled adequately by Motivated by the recent developments in existing standard distributions are sometimes generating new distributions and the need for found to follow distributions of some continuous extension and generalizations to combinations of two or more random variables more complex situations, this research paper with known probability distributions. introduces and studies the Gamma-Rayleigh Distribution (GRD). The Gamma and Rayleigh Developments on generating new distributions to distributions are well-known survival model naturally occurring phenomena have led to distributions; however, there might be some a number of new distributions being defined and survival data situations in which the two studied. Some of the earlier works include those distributions may not fit so well. Combining of Marsaglia (1965), Press (1969), Basu and them might therefore yield better results. Since Lochner (1972), and Lee et al. (1979). The trend estimation of parameters of a newly generated has been on the increase in recent years due to distribution is fundamental to application, the increasing need for adequate distributions to method of maximum likelihood is used in this model some data arising in practice (Gupta and work to estimate parameters of GRD. The Kundu, 1999; Eugene et al., 2004; Famoye et al., distribution is then applied to two survival data 2005; Akinsete et al., 2008; Alzaatreh et al., sets. MATLAB R2011b has been used for 2013b, Adeleke et al., 2013; Akarawak et al., implementations. 2013 and Akarawak et al., 2015). Furthermore, several studies (Mudholkar and Srivastava, 1993;
130 Nigerian Journal of Basic and Applied Science (December, 2017), 25(2): 130-142
Methodology given as: This article is organized as follows: derivation g (v) f (v)[ln{1 F(v)}] 1[1 F(v)] 1 and presentation of aspects of the Gamma- V () Rayleigh Distribution; parameter estimation of (2.6) the Gamma-Rayleigh Distribution and study of where f(v) and F(v) are the pdf and cdf of any the simulation and applications of the random variable V. distribution.
2.2 Derivation of the Gamma-Rayleigh THE GAMMA-RAYLEIGH DISTRIBUTION Distribution (GRD) (GRD) 2.2.1 Derivation of the pdf and cdf of GRD Method: The Transformed-Transformer The pdf and cdf of the Gamma-Rayleigh Technique of generating New Family of distribution is derived in this section as a class Continuous Distributions of Gamma-V family of generalized distributions. The Transformed-Transformer method was Theorem 2.1: recently introduced by Alzaatreh et al. (2013a) Let the pdf of a gamma distribution be for generating families of continuous distributions. Let F(v) be the cumulative f (x) (x) 1 ex and V Rayleigh distribution function (cdf) of any random () variable V and p( y ) the probability density distributed random variable with pdf v2 v2 v 2 2 function (pdf) of a random variable Y defined f (v) e 2 , and cdf F(v) 1 e 2 . on 0,. The cdf of a generalized family of 2 distributions is given as Then the pdf of the Gamma-Rayleigh log(1F (v)) distribution is given by: G(v) p(y)dy , (2.1) 0 2 g (v) v2 1 exp(v2 ); v 0;, 0 The family of distributions defined by (2.1) is V () called the ‘Transformed-Transformer’ family (or , (2.7) the Y-V family), where the random variable Y is Where, () is the gamma function. being transformed by another random variable
V (the transformer) into a new random variable Proof: with the support . According to Alzaatreh The pdf of the Gamma-V family of distribution is et al. (2013a), the corresponding pdf of the given by: generalized distribution in (2.1) is given by f (v) g(v) p(log[1 F(v)]) (2.2) 1 F(v) , (2.8) h(v)p(log[1 F(v)]) (2.3) Let V follow the Rayleigh distribution with pdf h(v)p[H(v)], (2.4) Where h(v) is the hazard function and H(v) is , and cdf , the cumulative hazard function of V with the 2 cdfF(v), which makes g(v) as arising from a v 2 2 weighted hazard function. One of the Y-V such that 1 F(v) e . Then the pdf in (2.8) families defined by Alzaatreh et al. (2013a) is becomes: 2 2 1 2 1 the Gamma-V family, obtained by letting Y v v v v 2 2 2 2 2 2 follow the Gamma distribution with parameters gV (v) e lne e () 2 and and having the pdf, 1 y , f (y) y e ; y> 0; α , λ >0. (2.5) 2 2 v 2 1 v ( 1) () 2 v 2 ve2 e 2 , According to Alzaatreh (2013a), the pdf of the 2 2 () 2 Gamma-V generalized family of distribution is
131 Akarawak et al: The Gamma-Rayleigh Distribution and Applications to Survival Data………………
2 2 2 v2 v v v 2 1 u 2 1 2 2 2 where, (,v ) u e du is the 2 2 2 0 2 1 2 v e e e , (2 ) () lower incomplete gamma function and 2 v 1 u 2 2 1 2 2 () u e du is the complete gamma 0 2 v e , 2 () function. v2 2 2 1 2 2 2 v e , (2.9) Proof: 2 () We prove the result using the fact that v Replacing 2 by , (2.9) becomes: G(v) g(x)dx , 2 0 2 2 2 v 2 g (v) v2 1ev . G(v) x 2 1ex dx , (2.14) V () () 0 Any random variable V that has the probability Changing variable as in (2.11), the result density function given in (2.7) is said to have follows. the Gamma-Rayleigh distribution with shape parameter and scale parameter and 2.2.2 Relationship with Other written as V ~ GRD( , ). Distributions The Gamma-Rayleigh distribution has two Theorem 2.2 parameters and . The function given in (2.7) is a valid probability density function. Corollary 2.1: When = 1 the pdf of Gamma- Rayleigh distribution reduces to the pdf of the
Proof: 2 1 Rayleigh distribution with parameter . Required to prove that g(v)dv 1. 2 0 Preposition 2.1: The Gamma-Rayleigh 2 2 g(v)dv v2 1ev dv , (2.10) distribution is a special case of the generalized 0 () 0 gamma distribution when p = 2, d = 2 and a= 1 1/ 2 2 u 2 . Let, u v v , so that du Proof of Preposition 2.1: , (2.11) dv 1 1 The generalized gamma distribution (Stacy, 2 2 2 u 1962) has a pdf of the form: Then (2.10) reduces to: p 2 1 p d 1 x 1/ 2 x exp 2 u u du a d a g(v)dv e 1 1 0 0 () 2 2 f (x) ; x 0; a, p,d 0 2 u d , p 1 () u 1eu du 1. (2.12) . (2.15) () 0 () When p = 2, d = 2 and a= , (2.15) Theorem 2.3: reduces to the Gamma-Rayleigh pdf given in The cumulative distribution function (cdf) of the (2.7). Gamma-Rayleigh distribution is given by: Corollary 2.2: If the random variable X follows (,v2 ) the Gamma-Rayleigh distribution with G(v) ; (2.13) () parameters and , then the Random variable Y = X2 follows the gamma distribution with parameters and .
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Proof of Corollary 2.2: y 1ey 2 , (2.16) Let X ~ GRD ( , ) and Y X such that () 1 2 dx 1 where (2.16) is the pdf of the gamma x y . Then 1 . dy 2y 2 distribution with parameters and . dx 1 f (y) g(x) ; x y 2 , dy
Plots of the Pdf and Cdf of the Gamma-Rayleigh Distribution
1.8 Scale = 1,Shape = 0.2 1.6 Scale = 1,Shape = 0.5 Scale = 1,Shape = 1 1.4 Scale = 1,Shape = 2 Scale = 1,Shape = 5 1.2
1 f(x) 0.8
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0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x Figure 1: Pdf Plot of GRD for different values of the shape parameter
2 Scale = 0.2,Shape = 1 1.8 Scale = 0.5,Shape = 1 Scale = 1,Shape = 1 1.6 Scale = 2,Shape = 1 Scale = 5,Shape = 1 1.4
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1 f(x)
0.8
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0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x Figure 2: Pdf Plot of GRD for different values of the scale parameter
1 Scale = 1,Shape = 0.5 0.9 Scale = 1,Shape = 1 Scale = 1,Shape = 3 0.8
0.7
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0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x Figure 3: Cdf Plots for GRD
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From Figures 1 and 2, it is observed that as the 3 shape parameter varies from 0.5 to 5, the 2 3 Gamma-Rayleigh distribution increasingly u3 E(X ) 3 , resembles the normal distribution; the spread 2 () reduces with increasing scale parameter. 2 4 Furthermore, the distribution shows positive u4 E(X ) 2 . skewness for different values of the scale () parameter. The cdf plots shown in Figure 3 (2.21) increase from 0 to 1 as X increases. and, Var(X ) E(X 2 ) E(X )2 , 2 1 SOME PROPERTIES OF THE GAMMA- ( 1) 2 RAYLEIGH DISTRIBUTION Moments () ()2 Theorem 2.4 2 2 1 The rth non-central moment of a Gamma- () 2 Rayleigh random variable X is given by: . (2.22) r ()2 2 E(X r ) (2.17) r The standard deviation of X is given by: 2 () Std(X ) Var(X ) Proof: 2 2 2 2 1 E(X r ) x2 r1ex dx , (2.18) () 0 2 () . (2.23) A change of variable from x to u reduces (2.18) ()2 to: r 1 1 Coefficient of variation of X u 2 eu du , (2.19) r 0 Coefficient of variation of X is given by: 2 () 2 2 1 1 r () Std (X ) 2 2 () 2 CV , 2 r E(X ) () 1 2 () 2 , (2.2 From (2.17) the first four moments are given 2 below: 2 1 () 1 2 . (2.25) 2 1 u E(X ) , 1 1 2 2 () The coefficient of variation given in (2.25) is expressed in terms of only. 2 1 () u2 E(X ) , () () Skewness and Kurtosis (2.20) The skewness of a distribution is a measure of its departure from symmetry, while the kurtosis is a measure of its peakedness. The skewness and kurtosis of a distribution are respectively given by:
134 Nigerian Journal of Basic and Applied Science (December, 2017), 25(2): 130-142
3 3 EX 3 Where, 3 3 312 21 and Skewness = 3 3 , (2.26) 2 4 4 4 413 61 2 31 . (2.28) EX 4 Kurtosis = 4 . (2.27) 4 4
For the Gamma-Rayleigh distribution, 3 3 1 1 3 1 2 2 2 2 , 3 3 3 3 2 () 2 ()2 2 ()3 3 2 3 1 1 () 3() 1 2 2 2 2 3 . (2.29) 2 ()3 2 4 1 3 1 1 4 6( 1) 3 2 2 2 2 2 4 , 2() 2 ()2 2()3 2 ()4 2 4 2 2 1 3 1 1 () 2 4() 6()( 1) 3 2 2 2 2 , (2.30) 2 ()4 3 2 3 1 1 () 3() 1 2 2 2 2 3 2 3 3 () Then, Skewness 3 3 2 2 1 () 2 2 () 3 2 3 1 1 3 () 3() 1 2 3 2 2 2 2 () , (2.31) 3 3 2 3 2 2 () 2 1 () 2
135 Akarawak et al: The Gamma-Rayleigh Distribution and Applications to Survival Data………………
2 4 2 2 1 3 1 1 () 2 4() 6()( 1) 3 4 2 2 2 2 Kurtosis 4 2 ()4 3 2 2 1 () 2 2 () 2 4 2 1 3 1 1 () 2 4 () 2 6()( 1) 3 2 4 2 2 2 2 () 2 () 4 2 2 1 () 2
2 4 2 2 1 3 1 1 () 2 4() 6()( 1) 3 2 2 2 2 4 . (2.32) 2 2 1 () 2
Corllary 2.1: Moment Generating Function r The moment generating function (mgf) of the 2 For GRD random variable V, , Gamma-Rayleigh random variable V is given r r by: 2 () r therefore, the moment generating function of V r is given by: 2 t M (t) . (2.33) V r r r0 r! 2 () 2 t r M (t) . Proof: V r r0 r! The moment generating function of a 2 () continuous random variable V is defined by: tV Corollary 2.2: Cumulants MV (t) E(e ) , According to Staurt and Ord (1994), the etv g(v)dv , (2.34) 0 rthcumulant cr is given in terms of the rth non- 2 3 r r (tv) (tv) t central moment r as But, etv 1 tv ... 2! 3! r! t r t r r0 cr logr (2.37) , (2.35) r1 r! r0 r! Therefore, (2.34) becomes: Expanding the logarithm in (2.37) givesc in r r r t v M V (t) g(v)dv , terms of and using (2.20) the first two 0 r! r r0 cumulants of GRD are given as: r r t t v r g(v)dv r , (2.36) 0 r0 r! r0 r!
136 Nigerian Journal of Basic and Applied Science (December, 2017), 25(2): 130-142
r (,v 2 ) S(v) 1 F(v) 1 (2.39) 2 () c , 1 r The hazard function of the Gamma-Rayleigh 2 () distribution is given by: 2 2 2 1 g(v) 2 v 2 1ev () 2 v 2 1ev h(v) ( 1) 2 S(v) ()[() (,v 2 )] (,v 2 ) c2 . () ()2 (2.40) (2.38) where, (,v 2 ) u 1eu du is the 2.3.2 Survival and Hazard Functions v2 The survival function for the Gamma-Rayleigh upper incomplete gamma function such that 2 2 distribution is given by: (,v ) (,v ) () .
Plots of the Survival and Hazard Functions of GRD
1 Scale = 1,Shape = 0.5 0.9 Scale = 1,Shape = 1 Scale = 1,Shape = 3 0.8
0.7
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0.5 S(x)
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0 0 0.5 1 1.5 2 2.5 3 x Figure 4: Plots of GRD Survival Function
The plot of the survival function is a decreasing function of X.
9 Scale = 1,Shape = 0.2 8 Scale = 1,Shape = 1 Scale = 1,Shape = 3 7 Scale = 1,Shape = 5
6
5 h(x) 4
3
2
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0 0 0.5 1 1.5 2 2.5 3 3.5 4 x Figure 5: Plot of the Gamma-Rayleigh Hazard Functions
The plots show that GRD hazard function is an increasing function of time.
Estimation of the Parameters Of GRD The Method of Maximum Likelihood (ML) In this section, the method of maximum Estimation likelihood and method of moments estimation of For estimating an unknown parameter θ, the the shape and scale parameters of the Gamma- likelihood principle can be used to obtain the Rayleigh distribution (GRD) are presented. maximum likelihood estimator (MLE) ˆ (Bai and Fu, 1987). The definition of maximum likelihood estimator is presented below.
137 Akarawak et al: The Gamma-Rayleigh Distribution and Applications to Survival Data………………
L(1 , 2 ,, k ) Definition 3.1: Likelihood Function (Mood et 0 1 al., 1974) L( , ,, ) The likelihood function of n random variables 1 2 k 0 X1, X2, …, Xn is defined to be the joint density of 2 the n random variables, say f X ,X ,,X (x1, x2 ,, xn;) , which is considered 1 2 n L( , ,, ) to be a function of . In particular, if 1 2 k 0, k X1, X 2 ,, X n is random sample from the 2 L density f (x;) , then the likelihood function is Such that, 0; i 1,,k . ˆ 2 given by L() f (x1;) f (x1;) f (x1;) i . The logarithm of the likelihood function has the same maximum point as the likelihood function Definition 3.2: Maximum Likelihood and is easier to compute. Estimator (MLE) Let Derivation of the ML Estimators for the n Parameters of the Gamma-Rayleigh L() L(; x1,, xn ) f (x1,) f (x2 ,) f (xn ,) fDistribution(xi ;) Let therei1 be a random sample of independent be the likelihood function of the random random variables from GRD each having the variables X1, X 2 ,, X n and 2 2 1 2 ˆ h(X ,, X ) a function of the random pdf, gV (v) v exp(v ) , the 1 n () variables. If the value of ˆ given by likelihood function is given by: n h(x1,, xn ) maximizes L() , then 2 2 1 2 L(,;vi ) vi exp(vi ) , is a maximum likelihood i1 () estimator of . Under certain regularity (3.3) n conditions, the maximum likelihood estimator of n n 2 2 1 2 θ is obtained by solving the likelihood equation: vi exp( vi ), () dL() i1 i1 0, (3.1) Taking the log of the likelihood function gives: d n 2 L log L nlog 2 n log nlog () (2 1) log v v 2 such that 0 . i ˆ 2 i1 , If the likelihood function contains k parameters, log L n that is, if v2 , n 2 L(1,,k ) f (xi ,1,,k ) , (3.2) n v 0, (3.4) i1 log L n() then, the maximum likelihood estimators are the nlog 2 log v , random variables () ˆ h (X , X ,, X ),ˆ h (X , X ,, X ),,ˆ h (X , X ,, X ) 1 1 1 2 n 2 2 1 2 n k nlogk 1n2() 2n logv 0 , (3.5)
, whose values maximize L(1,,k ) . If the (3.4) and (3.5) will be solved simultaneously to regularity conditions are satisfied, the point obtain the estimates of and . This cannot where the likelihood is a maximum is a solution be done analytically, therefore a numerical set of the k equations: technique will be adopted. Consequently, properties of the maximum likelihood estimators could not be derived in this paper.
138 Nigerian Journal of Basic and Applied Science (December, 2017), 25(2): 130-142
Simulation Study and Application fever and time to drop out from an insurance In this section, simulated data from the policy. The aim is to illustrate how the Gamma-Rayleigh distribution are analyzed. The distribution can be applied to real life data. simulation was done to study the behaviour and All simulations, analysis and plots are done some properties of the new distribution. using programme codes written in MATLAB Simulations were done using probability integral Application R2011b. transform. The probability integral transform equates the cdf of the GRD random variable V RESULT OF SIMULATION STUDY ON GRD to uniform random variates, say u such that u The simulation for GRD was done for different ~U(0,1). This procedure allows the random values of and . The results are presented values of V (the quantiles) to be obtained. in Table 1. Also, the distribution is applied to two real life survival data on time to recover from typhoid
Table 1: Result of Simulation Study on Gamma-Rayleigh Distribution Model Parameters Mean StdDev Variance SE of Skewness Kurtosis CV Mean GRD(1,0.5) = 1; =0.5 1.2462 0.6537 0.4274 0.0092 0.5999 3.1374 0.5246 GRD(1,2) = 1; = 2 0.6276 0.3311 0.1096 0.0047 0.6125 3.1334 0.5276 GRD(1,5) = 1; = 5 0.3950 0.2068 0.0428 0.0029 0.6385 3.3093 0.5235 GRD(0.5,1) = 0.5; =1 0.5661 0.4312 0.1859 0.0061 1.0557 4.1373 0.7617 GRD(2,1) = 2; = 1 1.3308 0.4755 0.2261 0.0067 0.4103 3.0981 0.3573 GRD(5,1) = 5; = 1 2.1785 0.4864 0.2366 0.0069 0.2527 3.0152 0.2233
The simulation results in Table 4.1 shows that Data Description and Exploration the skewness reduces as increases. The The two survival data used in this study are distribution shows kurtosis that is not so high, secondary data on time to recover from typhoid with smaller values of resulting in higher fever of patients and time to drop out from an kurtosis. Also, the variance decreases with insurance policy. The typhoid recovery time increasing values of the scale parameter θ. The data were collected from patients’ case notes in distribution shows a coefficient of variation (CV) General Hospital, Gbagada, Lagos while the of less than one for all values of the parameter policy drop out time data were collected from an and therefore, can be classified as a member of Insurance company. Table 2 gives the the hypo-exponential (CV < 1) class of summary of statistics and histograms for the distributions. two sets of survival data.
APPLICATIONS TO REAL LIFE DATA DISCUSSION In this section, the new distribution is applied to The estimates of the parameters and two data sets to illustrate their adequacies in performance criterion of AIC (figures in bold) for fitting real life data. The criterion used to check fitting GRD to the two survival data sets are model adequacy and performance is the Akaike given in Table 4.3 along with other distributions. Information Criterion (AIC). AIC is given by: AIC From the results, GRD is the most adequate in = 2k – 2logL; where L is the maximized value of fitting the Policy drop-out time data; followed by the likelihood and k is the number of WRD, Gamma and then Weibull distribution. parameters in the model. A rule of thumb is that The typhoid recovery time data is most fitted by a better model should have a smaller AIC Gamma, closely followed by GRD, Weibull and (Kwok et al., 2008). then Rayleigh.
139 Akarawak et al: The Gamma-Rayleigh Distribution and Applications to Survival Data………………
The results show that the newly introduced the Rayleigh distribution with the Gamma distribution is adequate in fitting the two survival distribution is worthwhile. data and in some cases produce better fits. It actually performs better than the Rayleigh distribution in both cases. Therefore, combining
Table 2: Summary of Statistics for Survival Data Statistics N Mean Variance Std Dev. Skewness Kurtosis Coef. of Var. Recovery 150 4.52 5.52 2.12597 1.097 1.809 0.47035 Time Policy 87 5.25 1.121 1.059 0.916 2.328 0.20171 Drop Out
Figure 6: Histograms of Survival Data
Both the descriptive results and histograms indicate that the data is peaked and skewed to the right.
RESULTS Table 3: Results of Fits of Distributions to Survival Data Data Weibull Rayleigh Gamma Lognormal GRD Time to Recover *Sh=2.2545 - Sh=4.6846 µ=1.3980 Sh=1.3335 from Typhoid *Sc=5.1129 Sc=3.5299 Sc=0.9649 σ=0.4872 Sc=0.0536 637.7309 639.3577 628.0819 663.4589 632.3686 Time to drop out Sh=4.8280 - Sh=25.7668 1.6392 Sh=6.5359 of Policy Sc=5.6885 Sc=3.7882 Sc=0.2039 0.1996 Sc=0.2277 272.6087 354.2708 273.008 267.50 256.6209 N/B: Sh = Shape; Sc = Scale; Figures in bold are AIC
140 Nigerian Journal of Basic and Applied Science (December, 2017), 25(2): 130-142
Fitted Density Plots for All Distributions
0.25 0.2 0.18 0.18 0.16 0.2 0.16 0.14 0.14 0.12 0.15 0.12 0.1
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f(x)
f(x) f(x) 0.08 0.1 0.08
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0 0 0 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 x x x Gamma Weibull Rayleigh 0.2 0.25 GRD 0.18
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0.18 0.35 0.4
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f(x) f(x) 0.08 0.15 0.15 0.06 0.1 0.1 0.04
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0 0 0 2 3 4 5 6 7 8 9 10 11 12 2 3 4 5 6 7 8 9 10 11 12 2 4 6 8 10 12 14 x x x Rayleigh Weibull 0.4 Gamma
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0.2 f(x) 0.15 0.15
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Figure 8: Fitted Density Plots for Policy Drop Out Time Data
CONCLUSION like the probability density function, cumulative This research work focused on generation of a distribution function, rth non-central moments, new distribution involving gamma and Rayleigh expectation, variances and cumulants were distributions. The resulting two-parameter derived. Expressions for its survival and hazard Gamma-Rayleigh Distribution (GRD) was functions were also presented. Plots of the pdf obtained through the Transformed-Transformer for different values of parameters and method. Properties of the resulting distribution simulation studies revealed that the newly
141 Akarawak et al: The Gamma-Rayleigh Distribution and Applications to Survival Data……………… derived distribution is unimodal, peaked and Eugene, N., Lee, C. and Famoye, F. (2002). Beta- skewed. Furthermore, plots of the hazard normal distribution and its function reveal that the distributions can be applications.Communication in Statistics- used to model data with increasing hazard Theory and Methods, 31(4): 497 – 512. rates. Famoye, F., Lee, C. and Olugbenga, O. (2005).The Beta-Weibull distribution. Journal of
Statistical Theory and Applications, 4(2): In order to make the distribution applicable and 121-138. relevant, estimators of the parameters were Gupta, R. D. and Kundu, D. (1999). Generalized derived using the method of maximum Exponential Distribution. Australian and likelihood. The distribution was used to model New Zealand Journal of Statistics, 41: 173 two sets of survival data and results show that it – 188. is competitively adequate in fitting the survival Gupta, R. D. and Kundu, D. (2001). Exponentiated data compared to the Weibull, Rayleigh, Exponential Family: An Alternative to Lognormal and Gamma distributions. Gamma and Weibull Distributions. Biometrical Journal, 43(1): 319 – 326. REFERENCES Kwok, O.M., Underhill, A.T., Berry, J. W., Luo, W., Adeleke, I. A, Akarawak, E. E. E, and Okafor, R. O. Ellot, T.R. and Yoon, M. (2008). Analyzing (2013). Investigating the Distribution of the Longitudinal Data with Multilevel Models: Ratio of Independent Beta and Weibull An Example with Individuals Living with Random Variables. Journal of Mathematics Lower Extremity Intra-articular Fractures. and Technology, 4(1): 16-22. Reliability Psychology, 53(3): 370-386. Akarawak, E. E. E., Adeleke, I. A. and Okafor, R. O. Lee, R. Y., Holland, B. S., Flueck, J. A. (1979). (2013). The Weibull-Rayleigh Distribution Distribution of a Ratio of Correlated and its Properties. Journal of Engineering Gamma Randon Variables, SIAM Journal Research, 18(1): 56-67. of Applied Mathematics, 36:304 – 320. Akarawak, E. E. E., Adeleke, I. A. and Okafor, R. O. Marsaglia, G. (1965). Ratios of normal variables (2015). On the Distribution of the Ratio of and ratios of sums of uniform Independent Gamma and Rayleigh variables.Journal of the American Random Variables. Journal of Scientific Statistical Association, 60: 193–204. Research and Developments, 15(1): 54-63. Mood, A. M., Graybill, F. A. and Boes, D. C. (1974). Akinsete, A., Famoye, F. and Lee, C. (2008). The Introduction to the Theory of Statistics rd beta-Pareto distribution.Statistics, 42:547- (3 ed). McGraw Hill International Book 563. Company. Tokyo. Alzaatreh, A., Lee, C. and Famoye, F. (2013a). A Mudholkar,G.S. and Srivastava, D.K. (1993). New Method for Generating Families of Exponentiated Weibull family for analyzing Continuous Distributions.Metron: Bathtub Failure Data. IEEE Trans. Rel. 42: International Journal of Statistics, 71(1): 299-302. 63-79. Pal, M., Ali, M. M. and Woo, J. (2006). Alzaatreh, A., Famoye, F. and Lee, C. Exponentiated Weibull Distribution. (2013b).Weibull-Pareto Distribution and its Statistica, 66 (2): 139-148. Applications. Communication in Statistics: Press, S. J. (1969). The t ratio distribution. Journal Theory and Methods, 42(9): 1673-1691. of the American Statistical Association, 64: Bai, Z.D. and Fu, J.C. (1987). On the Maximum- 242–252. Likelihood Estimator for the Location Stacy, E.W. (1962). A generalization of the gamma Parameter of a Cauchy Distribution. The distribution. Annals of Mathematical Canadian Journal of Statistics. 15(2): 137- Statistics, 33: 1187-1192. 146. Stuart, A. and Ord, J. K. (1994). Kendall's Advanced Basu, A. P. and Lochner, R. H. (1972). On the Theory of Statistics: Distribution Theory, Distribution of the Ratio of Two Random Vol 1. London: Arnold. Variables Having the Generalized Life Distributions. Technometrics. Journal Storage, 13(2): 281-287.
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