DOCSLIB.ORG

SAMPLE MOMENTS 1.1. Moments About the Origin (Raw Moments

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
SAMPLE MOMENTS 1.1. Moments About the Origin (Raw Moments SAMPLE MOMENTS 1. POPULATION MOMENTS 1.1. Moments about the origin (raw moments). The rth moment about the origin of a random variable X, 0 r denoted by µr, is the expected value of X ; symbolically, 0 r µr =E(X ) = X xr f(x) (1) x for r = 0, 1, 2, . when X is discrete and 0 r µr =E(X ) ∞ (2) = Z xrf(x) dx −∞ when X is continuous. The rth moment about the origin is only defined if E[ Xr ] exists. A moment about 0 the origin is sometimes called a raw moment. Note that µ1 = E(X)=µX , the mean of the distribution of X, or simply the mean of X. The rth moment is sometimes written as function of θ where θ is a vector of parameters that characterize the distribution of X. If there is a sequence of random variables, X1,X2,...Xn, we will call the rth population moment of the 0 ith random variable µi, r and define it as 0 r µi,r = E (Xi ) (3) 1.2. Central moments. The rth moment about the mean of a random variable X, denoted by µr , is the r expected value of ( X − µX ) symbolically, r µr =E[(X − µX ) ] r (4) = X ( x − µX ) f(x) x for r = 0, 1, 2, . when X is discrete and r µr =E[(X − µX ) ] ∞ Z r (5) = (x − µX ) f(x) dx −∞ th r when X is continuous. The r moment about the mean is only defined if E[ (X - µX ) ] exists. The rth moment about the mean of a random variable X is sometimes called the rth central moment of X. The rth r central moment of X about a is defined as E[ (X - a) ]. If a = µX, we have the rth central moment of X about µX. Note that Date: December 7, 2005. 1 2 SAMPLE MOMENTS ∞ Z µ1 = E[X − µX ]= (x − µX ) f(x) dx =0 −∞ ∞ (6) 2 Z 2 2 µ2 = E[(X − µX ) ]= (x − µX ) f(x) dx = Var(X)=σ −∞ Also note that all odd moments of X around its mean are zero for symmetrical distributions, provided such moments exist. th If there is a sequence of random variables, X1,X2, ... Xn, we will call the r central population moment of the ith random variable µi,r and define it as r 0 r µi,r = E Xi − µi,1 (7) 0 When the variables are identically distributed, we will drop the i subscript and write µ r and µ r . 2. SAMPLE MOMENTS 2.1. Definitions. Assume there is a sequence of random variables, X1,X2,...Xn. The first sample moment, usually called the average is defined by 1 n X¯ = X X (8) n n i i =1 Corresponding to this statistic is its numerical value, x¯n, which is defined by 1 n x¯ = X x (9) n n i i =1 where xi represents the observed value of Xi. The rth sample moment for any t is defined by 1 n X¯ r = X Xr (10) n n i i =1 This too has a numerical counterpart given by 1 n x¯r = X xr (11) n n i i=1 2.2. Properties of Sample Moments. ¯ r 2.2.1. Expected value of Xn . Taking the expected value of equation 10 we obtain 1 n 1 n E X¯ r = EX¯ r = X EXr = X µ0 (12) n n n i n i,r i=1 i=1 If the X’s are identically distributed, then n 1 0 0 E X¯ r = EX¯ r = X µ = µ (13) n n n r r i=1 SAMPLE MOMENTS 3 ¯ r 2.2.2. Variance of Xn. First consider the case where we have a sample X1,X2, ... ,Xn. 1 n ! 1 n ! VarX¯ r = Var X Xr = Var X Xr (14) n n i n2 i i =1 i =1 If the X’s are independent, then 1 n VarX¯ r = X Var(Xr ) (15) n n2 i i =1 If the X’s are independent and identically distributed, then 1 VarX¯ r = Var(Xr ) (16) n n where X denotes any one of the random variables (because they are all identical). In the case where r =1, we obtain 2 ¯ 1 σ VarXn = Var( X )= (17) n n 3. SAMPLE CENTRAL MOMENTS 3.1. Definitions. Assume there is a sequence of random variables, X1,X2, ... ,Xn. We define the sample central moments as n r 1 0 r C = X Xi − µ ,r =1, 2, 3,..., n n i,1 i=1 n 1 1 0 ⇒ C = X Xi − µ (18) n n i,1 i=1 n 2 1 0 2 ⇒ C = X Xi − µ n n i,1 i=1 0 These are only defined if µ i,1 is known. 3.2. Properties of Sample Central Moments. r r 3.2.1. Expected value of Cn. The expected value of Cn is given by n n r 1 0 r 1 E (C )= X E Xi − µ = X µi,r (19) n n i,1 n i=1 i =1 The last equality follows from equation 7. If the Xi are identically distributed, then r E ( Cn )=µ r 1 (20) E Cn =0 4 SAMPLE MOMENTS r 3.2.2. Variance of C n. First consider the case where we have a sample X1,X2, ... ,Xn. n n r 1 0 r ! 1 0 r ! Var( C )=Var X Xi − µ = Var X Xi − µ (21) n n i,1 n2 i,1 i=1 i=1 If the X’s are independently distributed, then n r 1 0 r Var( C )= X VarXi − µ (22) n n2 i,1 i=1 If the X’s are independent and identically distributed, then 1 r Var( Cr )= Var ( X − µ0 ) (23) n n 1 where X denotes any one of the random variables (because they are all identical). In the case where r =1, we obtain 1 0 Var C1 = Var[ X − µ ] n n 1 1 = Var[ X − µ ] n (24) 1 = σ2 − 2 Cov[ X, µ]+Var[ µ ] n 1 = σ2 n 4. SAMPLE ABOUT THE AVERAGE 4.1. Definitions. Assume there is a sequence of random variables, X1,X2,...Xn. Define the rth sample moment about the average as n r 1 ¯ r M = X Xi − Xn ,r=1, 2, 3,..., (25) n n i=1 This is clearly a statistic of which we can compute a numerical value. We denote the numerical value by, r mn, and define it as 1 n mr = X ( x − x¯ )r (26) n n i n i =1 In the special case where r = 1 we have n 1 1 ¯ M = X Xi − Xn n n i =1 1 n (27) = X X − X¯ n i n i =1 ¯ ¯ = Xn − Xn =0 4.2. Properties of Sample Moments about the Average when r = 2. SAMPLE MOMENTS 5 r 2 4.2.1. Alternative ways to write Mn. We can write Mn in an alternative useful way by expanding the squared term and then simplifying as follows n r 1 r M = X Xi − X¯n n n i =1 n 2 1 ¯ 2 ⇒ M = X Xi − Xn n n i=1 n ! 1 2 ¯ ¯ 2 = X X − 2 XiXn + X n i n i =1 (28) n n ¯ n 1 2 2Xn 1 ¯ 2 = X Xi − X Xi + X Xn n n i =1 n i =1 i =1 1 n = X X2 − 2X¯ 2 + X¯ 2 n i n n i =1 1 n ! = X X 2 − X¯ 2 n i n i =1 2 r 4.2.2. Expected value of Mn. The expected value of Mn is then given by 1 " n # E M 2 = E X X 2 − E X¯ 2 n n i n i =1 n 1 2 ¯ 2 ¯ = X E X − E Xn − Var(Xn) n i (29) i=1 2 1 n 1 n ! = X µ0 − X µ0 − Var(X¯ ) n i,2 n i,1 n i =1 i =1 The second line follows from the alternative definition of variance 2 Var( X )=E X 2 − [ E ( X )] 2 ⇒ E X 2 =[ E ( X )] + Var( X ) (30) ¯ 2 ¯ 2 ¯ ⇒ E Xn = E Xn + Var(Xn ) and the third line follows from equation 12. If the Xi are independent and identically distributed, then 6 SAMPLE MOMENTS 1 " n # E M 2 = E X X2 − E X¯ 2 n n i n i=1 2 1 n 1 n ! = X µ0 − X µ0 − Var(X¯ ) n i,2 n i,1 n i=1 i=1 (31) σ2 = µ0 − (µ0 )2 − 2 1 n 1 = σ2 − σ2 n n − 1 = σ2 n 0 0 where µ1 and µ2 are the first and second population moments, and µ2 is the second central population moment for the identically distributed variables. Note that this obviously implies " n # ¯ 2 2 E X Xi − X = nEMn i=1 n − 1 (32) = n σ2 n =(n − 1) σ2 2 4.2.3. Variance of Mn. By definition, 2 h 2 2i 2 2 VarMn = E Mn − EMn (33) The second term on the right on equation 33 is easily obtained by squaring the result in equation 31. n − 1 E M 2 = σ2 n n (34) 2 2 2 (n − 1) ⇒ E M 2 = EM2 = σ4 n n n2 Now consider the first term on the right hand side of equation 33. Write it as 2 n ! 2 2 1 ¯ 2 E hM i =E X Xi − Xn (35) n n i =1 1 n − ¯ 2 Now consider writing n Pi =1 Xi Xn as follows SAMPLE MOMENTS 7 n n 1 ¯ 2 1 ¯ 2 X Xi − X = X (Xi − µ) − (X − µ) n n i =1 i =1 n 1 2 = X Yi − Y¯ n (36) i =1 where Yi =Xi − µ Y¯ =X¯ − µ Obviously, n n ¯ 2 ¯ 2 ¯ ¯ X Xi − X = X Yi − Y , where Yi = Xi − µ,Y = X − µ (37) i =1 i =1 Now consider the properties of the random variable Yi which is a transformation of Xi.
Load more

Recommended publications
  • Use of Proc Iml to Calculate L-Moments for the Univariate Distributional Shape Parameters Skewness and Kurtosis
    Statistics 573 USE OF PROC IML TO CALCULATE L-MOMENTS FOR THE UNIVARIATE DISTRIBUTIONAL SHAPE PARAMETERS SKEWNESS AND KURTOSIS Michael A. Walega Berlex Laboratories, Wayne, New Jersey Introduction Exploratory data analysis statistics, such as those Gaussian. Bickel (1988) and Van Oer Laan and generated by the sp,ge procedure PROC Verdooren (1987) discuss the concept of robustness UNIVARIATE (1990), are useful tools to characterize and how it pertains to the assumption of normality. the underlying distribution of data prior to more rigorous statistical analyses. Assessment of the As discussed by Glass et al. (1972), incorrect distributional shape of data is usually accomplished conclusions may be reached when the normality by careful examination of the values of the third and assumption is not valid, especially when one-tail tests fourth central moments, skewness and kurtosis. are employed or the sample size or significance level However, when the sample size is small or the are very small. Hopkins and Weeks (1990) also underlying distribution is non-normal, the information discuss the effects of highly non-normal data on obtained from the sample skewness and kurtosis can hypothesis testing of variances. Thus, it is apparent be misleading. that examination of the skewness (departure from symmetry) and kurtosis (deviation from a normal One alternative to the central moment shape statistics curve) is an important component of exploratory data is the use of linear combinations of order statistics (L­ analyses. moments) to examine the distributional shape characteristics of data. L-moments have several Various methods to estimate skewness and kurtosis theoretical advantages over the central moment have been proposed (MacGillivray and Salanela, shape statistics: Characterization of a wider range of 1988).
    [Show full text]
  • Probability and Statistics
    APPENDIX A Probability and Statistics The basics of probability and statistics are presented in this appendix, provid- ing the reader with a reference source for the main body of this book without interrupting the flow of argumentation there with unnecessary excursions. Only the basic definitions and results have been included in this appendix. More complicated concepts (such as stochastic processes, Ito Calculus, Girsanov Theorem, etc.) are discussed directly in the main body of the book as the need arises. A.1 PROBABILITY, EXPECTATION, AND VARIANCE A variable whose value is dependent on random events, is referred to as a random variable or a random number. An example of a random variable is the number observed when throwing a dice. The distribution of this ran- dom number is an example of a discrete distribution. A random number is discretely distributed if it can take on only discrete values such as whole numbers, for instance. A random variable has a continuous distribution if it can assume arbitrary values in some interval (which can by all means be the infinite interval from −∞ to ∞). Intuitively, the values which can be assumed by the random variable lie together densely, i.e., arbitrarily close to one another. An example of a discrete distribution on the interval from 0 to ∞ (infinity) might be a random variable which could take on the values 0.00, 0.01, 0.02, 0.03, ..., 99.98, 99.99, 100.00, 100.01, 100.02, ..., etc., like a Bund future with a tick-size of 0.01% corresponding to a value change of 10 euros per tick on a nominal of 100,000 euros.
    [Show full text]
  • On Moments of Folded and Truncated Multivariate Normal Distributions
    On Moments of Folded and Truncated Multivariate Normal Distributions Raymond Kan Rotman School of Management, University of Toronto 105 St. George Street, Toronto, Ontario M5S 3E6, Canada (E-mail: [email protected]) and Cesare Robotti Terry College of Business, University of Georgia 310 Herty Drive, Athens, Georgia 30602, USA (E-mail: [email protected]) April 3, 2017 Abstract Recurrence relations for integrals that involve the density of multivariate normal distributions are developed. These recursions allow fast computation of the moments of folded and truncated multivariate normal distributions. Besides being numerically efficient, the proposed recursions also allow us to obtain explicit expressions of low order moments of folded and truncated multivariate normal distributions. Keywords: Multivariate normal distribution; Folded normal distribution; Truncated normal distribution. 1 Introduction T Suppose X = (X1;:::;Xn) follows a multivariate normal distribution with mean µ and k kn positive definite covariance matrix Σ. We are interested in computing E(jX1j 1 · · · jXnj ) k1 kn and E(X1 ··· Xn j ai < Xi < bi; i = 1; : : : ; n) for nonnegative integer values ki = 0; 1; 2;:::. The first expression is the moment of a folded multivariate normal distribution T jXj = (jX1j;:::; jXnj) . The second expression is the moment of a truncated multivariate normal distribution, with Xi truncated at the lower limit ai and upper limit bi. In the second expression, some of the ai's can be −∞ and some of the bi's can be 1. When all the bi's are 1, the distribution is called the lower truncated multivariate normal, and when all the ai's are −∞, the distribution is called the upper truncated multivariate normal.
    [Show full text]
  • 1. How Different Is the T Distribution from the Normal?
    Statistics 101–106 Lecture 7 (20 October 98) c David Pollard Page 1 Read M&M §7.1 and §7.2, ignoring starred parts. Reread M&M §3.2. The eects of estimated variances on normal approximations. t-distributions. Comparison of two means: pooling of estimates of variances, or paired observations. In Lecture 6, when discussing comparison of two Binomial proportions, I was content to estimate unknown variances when calculating statistics that were to be treated as approximately normally distributed. You might have worried about the effect of variability of the estimate. W. S. Gosset (“Student”) considered a similar problem in a very famous 1908 paper, where the role of Student’s t-distribution was first recognized. Gosset discovered that the effect of estimated variances could be described exactly in a simplified problem where n independent observations X1,...,Xn are taken from (, ) = ( + ...+ )/ a normal√ distribution, N . The sample mean, X X1 Xn n has a N(, / n) distribution. The random variable X Z = √ / n 2 2 Phas a standard normal distribution. If we estimate by the sample variance, s = ( )2/( ) i Xi X n 1 , then the resulting statistic, X T = √ s/ n no longer has a normal distribution. It has a t-distribution on n 1 degrees of freedom. Remark. I have written T , instead of the t used by M&M page 505. I find it causes confusion that t refers to both the name of the statistic and the name of its distribution. As you will soon see, the estimation of the variance has the effect of spreading out the distribution a little beyond what it would be if were used.
    [Show full text]
  • A Study of Non-Central Skew T Distributions and Their Applications in Data Analysis and Change Point Detection
    A STUDY OF NON-CENTRAL SKEW T DISTRIBUTIONS AND THEIR APPLICATIONS IN DATA ANALYSIS AND CHANGE POINT DETECTION Abeer M. Hasan A Dissertation Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY August 2013 Committee: Arjun K. Gupta, Co-advisor Wei Ning, Advisor Mark Earley, Graduate Faculty Representative Junfeng Shang. Copyright c August 2013 Abeer M. Hasan All rights reserved iii ABSTRACT Arjun K. Gupta, Co-advisor Wei Ning, Advisor Over the past three decades there has been a growing interest in searching for distribution families that are suitable to analyze skewed data with excess kurtosis. The search started by numerous papers on the skew normal distribution. Multivariate t distributions started to catch attention shortly after the development of the multivariate skew normal distribution. Many researchers proposed alternative methods to generalize the univariate t distribution to the multivariate case. Recently, skew t distribution started to become popular in research. Skew t distributions provide more flexibility and better ability to accommodate long-tailed data than skew normal distributions. In this dissertation, a new non-central skew t distribution is studied and its theoretical properties are explored. Applications of the proposed non-central skew t distribution in data analysis and model comparisons are studied. An extension of our distribution to the multivariate case is presented and properties of the multivariate non-central skew t distri- bution are discussed. We also discuss the distribution of quadratic forms of the non-central skew t distribution. In the last chapter, the change point problem of the non-central skew t distribution is discussed under different settings.
    [Show full text]
  • On the Meaning and Use of Kurtosis
    Psychological Methods Copyright 1997 by the American Psychological Association, Inc. 1997, Vol. 2, No. 3,292-307 1082-989X/97/$3.00 On the Meaning and Use of Kurtosis Lawrence T. DeCarlo Fordham University For symmetric unimodal distributions, positive kurtosis indicates heavy tails and peakedness relative to the normal distribution, whereas negative kurtosis indicates light tails and flatness. Many textbooks, however, describe or illustrate kurtosis incompletely or incorrectly. In this article, kurtosis is illustrated with well-known distributions, and aspects of its interpretation and misinterpretation are discussed. The role of kurtosis in testing univariate and multivariate normality; as a measure of departures from normality; in issues of robustness, outliers, and bimodality; in generalized tests and estimators, as well as limitations of and alternatives to the kurtosis measure [32, are discussed. It is typically noted in introductory statistics standard deviation. The normal distribution has a kur- courses that distributions can be characterized in tosis of 3, and 132 - 3 is often used so that the refer- terms of central tendency, variability, and shape. With ence normal distribution has a kurtosis of zero (132 - respect to shape, virtually every textbook defines and 3 is sometimes denoted as Y2)- A sample counterpart illustrates skewness. On the other hand, another as- to 132 can be obtained by replacing the population pect of shape, which is kurtosis, is either not discussed moments with the sample moments, which gives or, worse yet, is often described or illustrated incor- rectly. Kurtosis is also frequently not reported in re- ~(X i -- S)4/n search articles, in spite of the fact that virtually every b2 (•(X i - ~')2/n)2' statistical package provides a measure of kurtosis.
    [Show full text]
  • Stochastic Time-Series Spectroscopy John Scoville1 San Jose State University, Dept
    Stochastic Time-Series Spectroscopy John Scoville1 San Jose State University, Dept. of Physics, San Jose, CA 95192-0106, USA Spectroscopically measuring low levels of non-equilibrium phenomena (e.g. emission in the presence of a large thermal background) can be problematic due to an unfavorable signal-to-noise ratio. An approach is presented to use time-series spectroscopy to separate non-equilibrium quantities from slowly varying equilibria. A stochastic process associated with the non-equilibrium part of the spectrum is characterized in terms of its central moments or cumulants, which may vary over time. This parameterization encodes information about the non-equilibrium behavior of the system. Stochastic time-series spectroscopy (STSS) can be implemented at very little expense in many settings since a series of scans are typically recorded in order to generate a low-noise averaged spectrum. Higher moments or cumulants may be readily calculated from this series, enabling the observation of quantities that would be difficult or impossible to determine from an average spectrum or from prinicipal components analysis (PCA). This method is more scalable than PCA, having linear time complexity, yet it can produce comparable or superior results, as shown in example applications. One example compares an STSS-derived CO2 bending mode to a standard reference spectrum and the result of PCA. A second example shows that STSS can reveal conditions of stress in rocks, a scenario where traditional methods such as PCA are inadequate. This allows spectral lines and non-equilibrium behavior to be precisely resolved. A relationship between 2nd order STSS and a time-varying form of PCA is considered.
    [Show full text]
  • The Normal Moment Generating Function
    MSc. Econ: MATHEMATICAL STATISTICS, 1996 The Moment Generating Function of the Normal Distribution Recall that the probability density function of a normally distributed random variable x with a mean of E(x)=and a variance of V (x)=2 is 2 1 1 (x)2/2 (1) N(x; , )=p e 2 . (22) Our object is to nd the moment generating function which corresponds to this distribution. To begin, let us consider the case where = 0 and 2 =1. Then we have a standard normal, denoted by N(z;0,1), and the corresponding moment generating function is dened by Z zt zt 1 1 z2 Mz(t)=E(e )= e √ e 2 dz (2) 2 1 t2 = e 2 . To demonstate this result, the exponential terms may be gathered and rear- ranged to give exp zt exp 1 z2 = exp 1 z2 + zt (3) 2 2 1 2 1 2 = exp 2 (z t) exp 2 t . Then Z 1t2 1 1(zt)2 Mz(t)=e2 √ e 2 dz (4) 2 1 t2 = e 2 , where the nal equality follows from the fact that the expression under the integral is the N(z; = t, 2 = 1) probability density function which integrates to unity. Now consider the moment generating function of the Normal N(x; , 2) distribution when and 2 have arbitrary values. This is given by Z xt xt 1 1 (x)2/2 (5) Mx(t)=E(e )= e p e 2 dx (22) Dene x (6) z = , which implies x = z + . 1 MSc. Econ: MATHEMATICAL STATISTICS: BRIEF NOTES, 1996 Then, using the change-of-variable technique, we get Z 1 1 2 dx t zt p 2 z Mx(t)=e e e dz 2 dz Z (2 ) (7) t zt 1 1 z2 = e e √ e 2 dz 2 t 1 2t2 = e e 2 , Here, to establish the rst equality, we have used dx/dz = .
    [Show full text]
  • Unbiased Central Moment Estimates
    Unbiased Central Moment Estimates Inna Gerlovina1;2, Alan Hubbard1 February 11, 2020 1University of California, Berkeley 2University of California, San Francisco [email protected] Contents 1 Introduction 2 2 Estimators: na¨ıve biased to unbiased 2 3 Estimates obtained from a sample 4 4 Higher order estimates (symbolic expressions for expectations) 5 1 1 Introduction Umoments package calculates unbiased central moment estimates and their two-sample analogs, also known as pooled estimates, up to sixth order (sample variance and pooled variance are commonly used second order examples of corresponding estimators). Orders four and higher include powers and products of moments - for example, fifth moment and a product of second and third moments are both of fifth order; those estimators are also included in the package. The estimates can be obtained directly from samples or from na¨ıve biased moment estimates. If the estimates of orders beyond sixth are needed, they can be generated using the set of functions also provided in the package. Those functions generate symbolic expressions for expectations of moment combinations of an arbitrary order and, in addition to moment estimation, can be used for other derivations that require such expectations (e.g. Edgeworth expansions). 2 Estimators: na¨ıve biased to unbiased For a sample X1;:::;Xn, a na¨ıve biased estimate of a k'th central moment µk is n 1 X m = (X − X¯)k; k = 2; 3;:::: (1) k n i i=1 These biased estimates together with the sample size n can be used to calculate unbiased n estimates of a given order, e.g.
    [Show full text]
  • Volatility Modeling Using the Student's T Distribution
    Volatility Modeling Using the Student’s t Distribution Maria S. Heracleous Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Economics Aris Spanos, Chair Richard Ashley Raman Kumar Anya McGuirk Dennis Yang August 29, 2003 Blacksburg, Virginia Keywords: Student’s t Distribution, Multivariate GARCH, VAR, Exchange Rates Copyright 2003, Maria S. Heracleous Volatility Modeling Using the Student’s t Distribution Maria S. Heracleous (ABSTRACT) Over the last twenty years or so the Dynamic Volatility literature has produced a wealth of uni- variateandmultivariateGARCHtypemodels.Whiletheunivariatemodelshavebeenrelatively successful in empirical studies, they suffer from a number of weaknesses, such as unverifiable param- eter restrictions, existence of moment conditions and the retention of Normality. These problems are naturally more acute in the multivariate GARCH type models, which in addition have the problem of overparameterization. This dissertation uses the Student’s t distribution and follows the Probabilistic Reduction (PR) methodology to modify and extend the univariate and multivariate volatility models viewed as alternative to the GARCH models. Its most important advantage is that it gives rise to internally consistent statistical models that do not require ad hoc parameter restrictions unlike the GARCH formulations. Chapters 1 and 2 provide an overview of my dissertation and recent developments in the volatil- ity literature. In Chapter 3 we provide an empirical illustration of the PR approach for modeling univariate volatility. Estimation results suggest that the Student’s t AR model is a parsimonious and statistically adequate representation of exchange rate returns and Dow Jones returns data.
    [Show full text]
  • The Multivariate Normal Distribution=1See Last Slide For
    Moment-generating Functions Definition Properties χ2 and t distributions The Multivariate Normal Distribution1 STA 302 Fall 2017 1See last slide for copyright information. 1 / 40 Moment-generating Functions Definition Properties χ2 and t distributions Overview 1 Moment-generating Functions 2 Definition 3 Properties 4 χ2 and t distributions 2 / 40 Moment-generating Functions Definition Properties χ2 and t distributions Joint moment-generating function Of a p-dimensional random vector x t0x Mx(t) = E e x t +x t +x t For example, M (t1; t2; t3) = E e 1 1 2 2 3 3 (x1;x2;x3) Just write M(t) if there is no ambiguity. Section 4.3 of Linear models in statistics has some material on moment-generating functions (optional). 3 / 40 Moment-generating Functions Definition Properties χ2 and t distributions Uniqueness Proof omitted Joint moment-generating functions correspond uniquely to joint probability distributions. M(t) is a function of F (x). Step One: f(x) = @ ··· @ F (x). @x1 @xp @ @ R x2 R x1 For example, f(y1; y2) dy1dy2 @x1 @x2 −∞ −∞ 0 Step Two: M(t) = R ··· R et xf(x) dx Could write M(t) = g (F (x)). Uniqueness says the function g is one-to-one, so that F (x) = g−1 (M(t)). 4 / 40 Moment-generating Functions Definition Properties χ2 and t distributions g−1 (M(t)) = F (x) A two-variable example g−1 (M(t)) = F (x) −1 R 1 R 1 x1t1+x2t2 R x2 R x1 g −∞ −∞ e f(x1; x2) dx1dx2 = −∞ −∞ f(y1; y2) dy1dy2 5 / 40 Moment-generating Functions Definition Properties χ2 and t distributions Theorem Two random vectors x1 and x2 are independent if and only if the moment-generating function of their joint distribution is the product of their moment-generating functions.
    [Show full text]
  • Expectation and Functions of Random Variables
    POL 571: Expectation and Functions of Random Variables Kosuke Imai Department of Politics, Princeton University March 10, 2006 1 Expectation and Independence To gain further insights about the behavior of random variables, we first consider their expectation, which is also called mean value or expected value. The definition of expectation follows our intuition. Definition 1 Let X be a random variable and g be any function. 1. If X is discrete, then the expectation of g(X) is defined as, then X E[g(X)] = g(x)f(x), x∈X where f is the probability mass function of X and X is the support of X. 2. If X is continuous, then the expectation of g(X) is defined as, Z ∞ E[g(X)] = g(x)f(x) dx, −∞ where f is the probability density function of X. If E(X) = −∞ or E(X) = ∞ (i.e., E(|X|) = ∞), then we say the expectation E(X) does not exist. One sometimes write EX to emphasize that the expectation is taken with respect to a particular random variable X. For a continuous random variable, the expectation is sometimes written as, Z x E[g(X)] = g(x) d F (x). −∞ where F (x) is the distribution function of X. The expectation operator has inherits its properties from those of summation and integral. In particular, the following theorem shows that expectation preserves the inequality and is a linear operator. Theorem 1 (Expectation) Let X and Y be random variables with finite expectations. 1. If g(x) ≥ h(x) for all x ∈ R, then E[g(X)] ≥ E[h(X)].
    [Show full text]