Let π1,  π2, ... π be k (>_2) populations. Let  πi (i = 1, 2, ..., k) be characterized by the uniform distributionon (ai, bi), where exactly one of ai and bi is unknown. With unequal sample sizes, suppose that we wish to select arandom-size subset of the populations containing the one withthe smallest value of 0i = bi - ai. Rule Ri selects πi iff a likelihood-based k-dimensional confidence region for the unknown (01,..., 0k) contains at least one point having 0i as its smallest component. A second rule, R, is derived through a likelihood ratio and is equivalent to that of Barr and Rizvi (1966) when the sample sizes are equal. Numerical comparisons are made. The results apply to the larger class of densities g(z; 0i) = M(z)Q(0i) iff a(0i) < z < b(0i). Extensions to the cases when both ai and bi are unknown and when 0max is of interest are i i indicated. / digitalisering@umu

Let π1, ..., πk be k(&gt;_2) populations such that πi, i = 1, 2, ..., k, is characterized by the normal distribution with unknown mean and ui variance aio2 , where ai is known and o2 may be unknown. Suppose that on the basis of independent samples of size ni from π (i=1,2,...,k), we are interested in selecting a random-size subset of the given populations which hopefully contains the population with the largest mean.Based on likelihood ratios, several new procedures for this problem are derived in this report. Some of these procedures are compared with the classical procedure of Gupta (1956,1965) and are shown to be better in certain respects. / <p>Ny rev. utg.</p><p>This is a slightly revised version of Statistical Research Report No. 1978-6.</p> / digitalisering@umu

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(&gt; 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) &gt;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&lt;j&lt;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. / digitalisering@umu