A gamma goodness-of-fit test based on characteristic independence of the mean and coefficient of variation

Abstract

Characterization theorems in probability and statistics are widely appreciated for their role in clarifying the structure of the families of probability distributions. Less well known is the role characterization theorems have as a natural, logical and effective starting point for constructing goodness-of-fit tests. The characteristic independence of the mean and variance and of the mean and the third central moment of a normal sample were used, respectively, by Lin and Mudholkar [1980. A simple test for normality against asymmetric alternatives. Biometrika 67, 455–461] and by Mudholkar et al. [2002a. Independence characterizations and testing normality against skewness-kurtosis alternatives. J. Statist. Plann. Inference 104, 485–501] for developing tests of normality. The characteristic independence of the maximum likelihood estimates of the population parameters was similarly used by Mudholkar et al. [2002b. Independence characterization and inverse Gaussian goodness-of-fit. Sankhya A 63, 362–374] to develop a test of the composite inverse Gaussian hypothesis. The gamma models are extensively used for applied research in the areas of econometrics, engineering and biomedical sciences; but there are few goodness-of-fit tests available to test if the data indeed come from a gamma population. In this paper we employ Hwang and Hu's [1999. On a characterization of the gamma distribution: the independence of the sample mean and the sample coefficient of variation. Ann. Inst. Statist. Math. 51, 749–753] characterization of the gamma population in terms of the independence of sample mean and coefficient of variation for developing such a test. The asymptotic null distribution of the proposed test statistic is obtained and empirically refined for use with samples of moderate size.