Tests and confidence intervals for the shape parameter of a gamma distribution

Abstract

Inference based on the Central Limit Theorem has only first order accuracy. We give tests and confidence intervals (CIs) of second orderaccuracy for the shape parameter ρ of a gamma distribution for both the unscaled and scaled cases. Tests and CIs based on moment and cumulant estimates are considered as well as those based on the maximum likelihood estimate (MLE). For the unscaled case the MLE is the moment estimate of order zero; the most efficient moment estimate of integral order is the sample mean, having asymptotic relative efficiency (ARE) .61 when ρ= 1. For the scaled case the most efficient moment estimate is a functionof the mean and variance. Its ARE is .39 when ρ = 1. Our motivation for constructing these tests of ρ = 1 and CIs forρ is to provide a simple and convenient method for testing whether a distribution is exponential in situations such as rainfall models where such an assumption is commonly made.