Likelihood Ratio Procedures for Subset Selection and Ranking Problems

This report deals with procedures for random-size subset selection from k(> 2) given populations where the distribution of ir^(i = l, ..., k) has a density f^(x;0^). Let ••• -®[k] denote unknown values of the parameters, and let ^[i]» ***'ïï[k] denote the corresponding populations. First, we have considered the problem of selection for consider the /s procedure that selects TT. if sup L(0;x) > c L(0;x), where L(*;x) is the 1 e e u . i total likelihood function, where is the region m the parameter space for i A 9= (0̂, ..., 0 )̂ having 0^ as the largest component, where 9 is the maxi­ m u m l i k e l i h o o d e s t i m a t e o f 0 , a n d w h e r e c i s a g i v e n c o n s t a n t w i t h 0 < c < l . With the densities satisfying seme reasonable requirements given in this re­ port, we have shown that for each i, the probability of including the selected subset is decreasing in ®[j] f°r j t i anc* increasing in We have then derived some results on selection for the t(> 1) best popula­ tions, thereby generalizing the results for t = 1. For this problem, we have considered a) selection of a set whose elements consist of subsets of the given populations having t members, and requiring that the set of the t • » • • • best populations is included with probability at least P , b) selection of a subset of the populations so as to include all the t best populations with probability at least P'*, and c) selection of a subset of the popula­ tions such that TT[j ̂ is included with probability at least P*, j=k-t+l, .•., k. In the final section, we have discussed the relation between the theories of subset selection based on likelihood ratios and statistical in­ ference under order restrictions, and have considered the complete ranking problem.