Interval Estimation and Hypothesis Testing on the Common Shape Parameter of Two Gamma Populations with Different and Unknown Scales

Abstract

Suppose there are two Gamma populations with a common shape and different scale parameters. The goal of the study is to derive confidence intervals and test the hypothesis for the associated model parameters. The asymptotic confidence intervals for the common shape and the two nuisance scale parameters are derived using the Fisher’s matrix. The parametric bootstrap intervals, namely, the boot-t and boot-p intervals are derived using the bootstrap sampling procedure. The highest posterior density intervals using Markov chain Monte Carlo procedure and some prior probabilities for the parameters are derived. All these intervals are obtained numerically, as no closed-form expressions are available. A numerical comparison among all the proposed intervals has been made using their average length and coverage probability. For hypothesis testing, several tests such as the Computational Approach Test (CAT), the Modified Computational Approach Test (MCAT), the Likelihood Ratio Test (LRT), the Parametric Bootstrap Likelihood Ratio Test (PBLRT), and the Standardized Likelihood Ratio Test (SLRT) have been proposed. Finally, the sizes and the powers of all the proposed tests have been computed using the Monte Carlo simulation procedure. Finally, a real-life data analysis has been carried out for the potential application of the proposed model.