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Some Generalized Gamma Distribution
Saralees Nadarajah,Arjun K. Gupta 한국통계학회 2007 Journal of the Korean Statistical Society Vol.36 No.1
Gamma distributions are some of the most popular models for hydro-logical processes. In this paper, a very exible family which contains thegamma distribution as a particular case is introduced. Evidence of exibil-ity is shown by examining the shape of its pdf and the associated hazardrate function. A comprehensive treatment of the mathematical propertiesis provided by deriving expressions for thenth moment, moment generatingfunction, characteristic function, Renyi entropy and the asymptotic distri-bution of the extreme order statistics. Estimation and simulation issues arealso considered. Finally, a detailed application to drought data from theState of Nebraska is illustrated.AMS 2000 subject classications.Primary 33C90; Secondary 62E99.Keywords.Drought modeling, gamma distribution, generalized gamma distribution.1. IntroductionA random variable X is said to have the standard gamma distribution if itsprobability density function (pdf) is given byf(x) =x1 exp( x)()(1.1)forx > 0, > 0 and > 0. Gamma distributions are some of the most popularmodels for hydrological processes (Yue, 2001; Yueet al., 2001; Shiauet al., 2006;references therein). The aim of this paper is to introduce a generalization of(1.1) that could have much wider applicability in hydrology. The generalizationis given by the pdff(x) = Cx1(x + z)exp( x) (1.2)Received May 2006; accepted September 2006.1Corresponding author. School of Mathematics, University of Manchester, Manchester M601QD, U.K. (e-mail: saralees.nadarajah@manchester.ac.uk)
THE BIVARIATE F<sub>3</sub>-BETA DISTRIBUTION
Nadarajah Saralees Korean Mathematical Society 2006 대한수학회논문집 Vol.21 No.2
A new bivariate beta distribution based on the Appell function of the third kind is introduced. Various representations are derived for its product moments, marginal densities, marginal moments, conditional densities and conditional moments. The method of maximum likelihood is used to derive the associated estimation procedure as well as the Fisher information matrix.
SOME ALGEBRA FOR GENERALIZED PLANCK RANDOM VARIABLES
Nadarajah, Saralees Korean Mathematical Society 2007 대한수학회논문집 Vol.22 No.3
The exact distributions of X + Y, XY and X/(X + Y) are derived when X and Y are independent generalized Planck random variables.
Some algebra for Pearson type VII random variables
Saralees Nadarajah 대한수학회 2008 대한수학회보 Vol.45 No.2
The distributions of products and ratios of random variablesare of interest in many areas of the sciences. In this paper, the exactdistributions of the product |XY| and the ratio |X/Y| are derived when X and Y are independent Pearson type VII random variables.
A Skewed Generalized t Distribution
Saralees Nadarajah 한국통계학회 2005 Journal of the Korean Statistical Society Vol.34 No.4
Skewed t distributions have attracted significant attention in the last few years. In this paper, a generalization – referred to as the skewed generalized t distribution – with the pdf f(x) = 2g(x)G(¸x) is introduced, where g(¢) and G(¢) are taken, respectively, to be the pdf and the cdf of the generalized t distribution due to McDonald and Newey (1984, 1988). Several particular cases of this distribution are identified and various representations for its moments derived. An application is provided to rainfall data from Orlando, Florida.
SOME ALGEBRA FOR PEARSON TYPE VII RANDOM VARIABLES
Nadarajah, Saralees Korean Mathematical Society 2008 대한수학회보 Vol.45 No.2
The distributions of products and ratios of random variables are of interest in many areas of the sciences. In this paper, the exact distributions of the product |XY| and the ratio |X/Y| are derived when X and Y are independent Pearson type VII random variables.
SOME POPULAR WAVELET DISTRIBUTION
Nadarajah, Saralees Korean Mathematical Society 2007 대한수학회보 Vol.44 No.2
The modern approach for wavelets imposes a Bayesian prior model on the wavelet coefficients to capture the sparseness of the wavelet expansion. The idea is to build flexible probability models for the marginal posterior densities of the wavelet coefficients. In this note, we derive exact expressions for a popular model for the marginal posterior density.
The model for fracture toughness
SAralees Nadarajah 대한기계학회 2008 JOURNAL OF MECHANICAL SCIENCE AND TECHNOLOGY Vol.22 No.7
It is pointed out that the distribution introduced by Neville [1] (for modeling fracture toughness to failures) is contained by at least two families of distributions known since the 1940s. Some elementary statistical properties of these families are discussed. Six data sets on fracture toughness are used to demonstrate that these families are much better models for fracture toughness than the one introduced by Neville [1].