Distribution of ratio of extreme of isotonic estimators of chi-square variables with applications

Let V 1 , … , V k be k independent chi-square variables each with ν degrees of freedom (d.f.) and let W 1 ⩽ ⋯ ⩽ W k be the corresponding isotonic estimators computed by the pool adjacent violator algorithm under simple ordering among Vs . In this paper, a recursive integration algorithm for computing the distribution function of statistic T = W k / W 1 is presented. The test statistic F max * proposed by Fujino (Biometrika 66 (1979) 133-139) to test the homogeneity of variances against ordered alternative follows same null distribution as of the statistic T. Fujino discussed the null F max * distribution of and provided critical constant only for k = 3 , … , 6 , which limited its applications. The recursive procedure given in this article can be implemented efficiently for sufficient higher value of k. Test procedures based on statistic T for testing homogeneity against simple ordered alternative of k normal variances and equality of scale parameters of k exponential distributions are discussed separately with tables of critical constants for their implementation. Correctness of the critical constants is verified through simulation study. The proposed methodology is also extended to address the problem of testing exponentiality (constant failure rate) versus increasing failure rate, using isotonic estimators of normalized spaces between consecutive order statistics, along with tables of relevant critical constants.