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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)
SOME GENERALIZED GAMMA DISTRIBUTION
Nadarajah Saralees,Gupta Arjun K. The Korean Statistical Society 2007 Journal of the Korean Statistical Society Vol.36 No.1
Gamma distributions are some of the most popular models for hydrological processes. In this paper, a very flexible family which contains the gamma distribution as a particular case is introduced. Evidence of flexibility is shown by examining the shape of its pdf and the associated hazard rate function. A comprehensive treatment of the mathematical properties is provided by deriving expressions for the nth moment, moment generating function, characteristic function, Renyi entropy and the asymptotic distribution of the extreme order statistics. Estimation and simulation issues are also considered. Finally, a detailed application to drought data from the State of Nebraska is illustrated.
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.
A SKEWED GENERALIZED t DISTRIBUTION
NADARAJAH SARALEES The Korean Statistical Society 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(${\lambda}x$) is introduced, where g(${\cdot}$) and G (${\cdot}$) 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.
On The Product of Laplace and Bessel Random Variables
Nadarajah, Saralees,Ali, M. Masoom Korean Data and Information Science Society 2004 한국데이터정보과학회지 Vol.15 No.4
The distribution of the product |XY| is derived when X and Y are Laplace and Bessel random variables distributed independently of each other.
ON THE PRODUCT OF t AND BESSEL RANDOM VARIABLES
NADARAJAH SARALEES Korean Mathematical Society 2005 대한수학회논문집 Vol.20 No.3
The distribution of products of random variables is of interest in many areas of the sciences, engineering and medicine. This has increased the need to have available the widest possible range of statistical results on products of random variables. In this note, the distribution of the product | XY | is derived when X and Y are Student's t and Bessel function random variables distributed independently of each other.
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.
THE BIVARIATE GAMMA EXPONENTIAL DISTRIBUTION WITH APPLICATION TO DROUGHT DATA
Nadarajah, Saralees 한국전산응용수학회 2007 Journal of applied mathematics & informatics Vol.24 No.1
The exponential and the gamma distributions have been the traditional models for drought duration and drought intensity data, respectively. However, it is often assumed that the drought duration and drought intensity are independent, which is not true in practice. In this paper, an application of the bivariate gamma exponential distribution is provided to drought data from Nebraska. The exact distributions of R=X+Y, P=XY and W=X/(X+Y) and the corresponding moment properties are derived when X and Y follow this bivariate distribution.
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.
Robust Optimization for Stability of I-Walls and Levee System Resting on Sandy Foundation
Nadarajah Ravichandran,Lei Wang,Parishad Rahbari 대한토목학회 2022 KSCE JOURNAL OF CIVIL ENGINEERING Vol.26 No.1
During Hurricane Katrina in 2005 and the events thereafter, failures of levees with I-walls caused extensive flooding and damage. The geological background in the New Orleans area and associated uncertainties contributed significantly to the failures. To increase the robustness of the I-walls and levee system and reduce the associated risk of failure, the uncertainties of the system must be incorporated into the design procedures, especially in a geological environment mainly composed of sand deposits. This paper presents a robust optimization procedure to identify optimal designs for the stability of an I-walls and levee system supported on sandy foundation soil in the face of flood hazards. The uncertainties associated with the I-walls and levee system, including the strength parameters of levee and foundation soils and the height of the floodwater behind the I-walls, were considered in a systematic manner. The wall embedded depth, levee crown width, and slope ratio of the levee in the landside were considered as the design parameters. For the robust optimization, the construction cost of the I-walls and levee system and the standard deviation of the failure probability were considered as the design objectives. Finally, the multi-objective optimization resulted in a set of acceptable designs that were presented in a graphical form called Pareto front, which is combined with the knee point concept to provide useful decision aids for selecting the most preferred design that meets both the economics and performance requirements.