Exact Inferences for a Gamma Distribution

Abstract

A random sample from a two-parameter gamma distribution is considered, and it is shown how exact inferences can be obtained for properties of the distribution. Kolmogorov tests based on the empirical cumulative-distribution function of the data are inverted to construct an exact confidence set for the two parameters. This can be used, for example, to construct exact confidence bands for the cumulative-distribution function of the gamma distribution, which also provide simultaneous inferences on the quantiles of the distribution. The exact confidence set can also provide confidence intervals for the individual parameters and for other functions of the parameters, such as the mean of the distribution. The new methodology is computationally straightforward and examples of its implementation are provided. Comparisons are made with standard approximate inference procedures that rely on asymptotic properties of maximum-likelihood estimates of the parameters and with a Bayesian simulation approach. The methodology can also be used to test whether a data set can be modeled with a gamma distribution and to test whether independent data sets can be modeled with a common gamma distribution.