On the exact distribution of the likelihood ratio test for testing the homogeneity of scale parameters of several two-parameter exponential distributions: complete and censored samples

We consider the problem of providing the exact distribution of the likelihood ratio test (LRT) statistic for testing the homogeneity of scale parameters of k≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ k \ge 2 $$\end{document} two-parameter exponential distributions. To this end, we apply the Millen inverse transform and the Jacobi polynomial expansion to the moments of LRT statistic. We consider the problem when the data are either complete or censored under the different kinds of Type II censoring, such as the Type II right, progressively Type II right, and double Type II censoring schemes. We also discuss the exact null distribution of the LRT when the data are censored under the Type I censoring scheme.