Selection and ranking procedures based on likelihood ratios

University dissertation from Umeå : Umeå universitet

Abstract: This thesis deals with random-size subset selection and ranking procedures• • • )|(derived through likelihood ratios, mainly in terms of the P -approach.Let IT , . .. , IT, be k(> 2) populations such that IR.(i = l, . . . , k) hasJ_ K. — 12the normal distribution with unknwon mean 0. and variance a.a , where a.i i i2 . . is known and a may be unknown; and that a random sample of size n^ istaken from . To begin with, we give procedure (with tables) whichselects IT. if sup L(0;x) >c SUD L(0;X), where SÎ is the parameter space1for 0 = (0-^, 0^) ; where (with c: ß) is the set of all 0 with0. = max 0.; where L(';x) is the likelihood function based on the total1sample; and where c is the largest constant that makes the rule satisfy theP'-condition. Then, we consider other likelihood ratios, with intuitivelyreasonable subspaces of ß, and derive several new rules. Comparisons amongsome of these rules and rule R of Gupta (1956, 1965) are made using differentcriteria; numerical for k=3, and a Monte-Carlo study for k=10.For the case when the populations have the uniform (0,0^) distributions,and we have unequal sample sizes, we consider selection for the populationwith min 0.. Comparisons with Barr and Rizvi (1966) are made. Generalizai<j<k Jtions are given.Rule R^ is generalized to densities satisfying some reasonable assumptions(mainly unimodality of the likelihood, and monotonicity of the likelihoodratio). An exponential class is considered, and the results are exemplifiedby the gamma density and the Laplace density. Extensions and generalizationsto cover the selection of the t best populations (using various requirements)are given. Finally, a discussion oil the complete ranking problem,and on the relation between subset selection based on likelihood ratios andstatistical inference under order restrictions, is given.