Testing Parameters of a Gamma Distribution for Small Samples

Abstract

The gamma distribution is relevant to numerous areas of application in the physical, environmental, and biological sciences. The focus of this paper is on testing the shape, scale, and mean of the gamma distribution. Testing the shape parameter of the gamma distribution is relevant to failure time modeling where it can be used to determine if the failure rate is constant, increasing, or decreasing. Testing the scale parameter is also relevant to problems in survival analysis, where when the shape parameter κ=1, the reciprocal of the scale parameter measures the hazard function. Finally, testing the mean of the gamma distribution allows us to determine if the average concentration of an environmental contaminant is higher, lower, or equivalent to a health-based standard. In this paper, we first derive new small sample-based tests and then via simulation, we study the Type I error rate and statistical power of these tests. Results of these simulation studies reveal that in terms of maintaining Type I error rate, the new tests perform extremely well as long as the shape parameter is not too small, and even then the results are only slightly conservative. We illustrate the new tests using three real datasets taken from the fields of engineering, medicine, and environmental science. This article has supplementary material online.

Keywords:  :

• Gamma distribution
• Log-Gamma
• Power function
• Wilson–Hilferty approximation