Small Sample Tests for Shape Parameters of Gamma Distributions

Abstract

The introduction of shape parameters into statistical distributions provided flexible models that produced better fit to experimental data. The Weibull and gamma families are prime examples wherein shape parameters produce more reliable statistical models than standard exponential models in lifetime studies. In the presence of many independent gamma populations, one may test equality (or homogeneity) of shape parameters. In this article, we develop two tests for testing shape parameters of gamma distributions using chi-square distributions, stochastic majorization, and Schur convexity. The first one tests hypotheses on the shape parameter of a single gamma distribution. We numerically examine the performance of this test and find that it controls Type I error rate for small samples. To compare shape parameters of a set of independent gamma populations, we develop a test that is unbiased in the sense of Schur convexity. These tests are motivated by the need to have simple, easy to use tests and accurate procedures in case of small samples. We illustrate the new tests using three real datasets taken from engineering and environmental science. In addition, we investigate the Bayes’ factor in this context and conclude that for small samples, the frequentist approach performs better than the Bayesian approach.

Keywords:

• Bayes’ factor
• Beta distribution
• Dirichlet distribution
• Majorization
• Schur convex functions

Mathematics Subject Classification:

• Primary 97K80
• Secondary 62F03

Acknowledgments

The authors thank Dr. James Webber of the Wadsworth Center of the New York State Department of Health for providing the Asbestos data and also for providing helpful comments.

Funding

This work was supported in part by grants from the National Institute of Health (R01 MH69353).