Small Sample Tests for Shape Parameter(s) of Gamma Distributions

• Activity Number: 77
• Type: Contributed
• Date/Time: Sunday, August 1, 2010 : 4:00 PM to 5:50 PM
• Sponsor: Section on Physical and Engineering Sciences
• Author(s): Dulal Bhaumik*+ and Jerome Keating and Robert D. Gibbons and Kush Kapur
• Companies: The University of Texas at San Antonio and University of Illinois at Chicago and University of Illinois at Chicago and University of Illinois at Chicago
• Keywords: Dirichlet distribution ; Beta distribution ; Majorization ; Schur convex ; Stochastic ordering

Abstract

The use of shape parameters in the scientific literature opened new areas of research and allowed one to use models that produce better fits to experimental data. The Weibull and gamma families are prime examples where shape parameters produce more reliable models than the exponential. In the presence of many gamma-populations, one may test equality of shape parameters across a collection of independent populations. In this paper we develop three non-trivial tests for testing shape parameters of gamma distributions using probability inequality, stochastic majorization and Schur convexity. The first two tests, test hypotheses on the shape parameter of a single gamma distribution. We investigate the performances of these tests and find that both tests control Type I error rate for small samples. To compare several gammas, we develop a test that is unbiased in the sense of Schur convexity.