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Selection and ranking procedures based on likelihood ratios
Umeå University, Faculty of Science and Technology, Mathematical statistics.
1979 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

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.

Place, publisher, year, edition, pages
Umeå: Umeå universitet , 1979. , 24 p.
Keyword [en]
Subset selection, likelihood ratio, hypothesis testing, order restrictions, normal distribution, uniform distribution, complete ranking, P* -condition, loss function
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:umu:diva-73690OAI: oai:DiVA.org:umu-73690DiVA: diva2:633195
Public defence
1979-06-01, Samhällsvetarhuset, hörsal D, Umeå universitet, Umeå, 09:15
Projects
digitalisering@umu
Available from: 2013-06-26 Created: 2013-06-26 Last updated: 2013-07-02Bibliographically approved
List of papers
1. Subset selection based on likelihood ratios: the normal means case
Open this publication in new window or tab >>Subset selection based on likelihood ratios: the normal means case
1979 (English)Report (Other academic)
Abstract [en]

Let π1, ..., πk be k(>_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.

Place, publisher, year, edition, pages
Umeå: Umeå universitet, 1979. 70 p.
Series
Statistical research report, ISSN 0348-0399 ; 6
Keyword
Subset selection, likelihood ratio, order restrictions, loss function, normal distribution
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:umu:diva-74925 (URN)
Projects
digitalisering@umu
Note

Ny rev. utg.

This is a slightly revised version of Statistical Research Report No. 1978-6.

Available from: 2013-07-02 Created: 2013-07-02 Last updated: 2013-07-02Bibliographically approved
2. Subset selection based on likelihood from uniform and related populations
Open this publication in new window or tab >>Subset selection based on likelihood from uniform and related populations
1979 (English)Report (Other academic)
Abstract [en]

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.

Place, publisher, year, edition, pages
Umeå: Umeå universitet, 1979. 28 p.
Series
Statistical research report, ISSN 0348-0399 ; 7
Keyword
Subset selection, likelihood ratio, order restrictions, uniform distribution
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:umu:diva-74924 (URN)
Projects
digitalisering@umu
Available from: 2013-07-02 Created: 2013-07-02 Last updated: 2013-07-02Bibliographically approved
3. Likelihood ratio procedures for subset selection and ranking problems
Open this publication in new window or tab >>Likelihood ratio procedures for subset selection and ranking problems
1979 (English)Report (Other academic)
Abstract [en]

This report deals with procedures for random-size subset selection fromk(> 2) given populations where the distribution of ir^(i = l, ..., k)has a density f^(x;0^). Let ••• -®[k] denote unknown values ofthe parameters, and let ^[i]» ***'ïï[k] denote the corresponding populations.First, we have considered the problem of selection for consider the/sprocedure that selects TT. if sup L(0;x) > c L(0;x), where L(*;x) is the1 e e u . - - - - -itotal likelihood function, where is the region m the parameter space foriA9= (0^, ..., 0^) having 0^ as the largest component, where 9 is the maximum likelihood estimate of 0 , and where c is a given constant with 0 < c < l .With the densities satisfying seme reasonable requirements given in this report,we have shown that for each i, the probability of includingthe selected subset is decreasing in ®[j] f°r j t i anc* increasing inWe have then derived some results on selection for the t(> 1) best populations,thereby generalizing the results for t = 1. For this problem, we haveconsidered a) selection of a set whose elements consist of subsets of thegiven populations having t members, and requiring that the set of the t• » • • •best populations is included with probability at least P , b) selection ofa subset of the populations so as to include all the t best populationswith probability at least P'*, and c) selection of a subset of the populationssuch 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 thetheories of subset selection based on likelihood ratios and statistical inferenceunder order restrictions, and have considered the complete rankingproblem.

Place, publisher, year, edition, pages
Umeå: Umeå universitet, 1979. 31 p.
Series
Statistical research report, ISSN 0348-0399 ; 8
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:umu:diva-73606 (URN)
Projects
digitalisering@umu
Note

There are some occurring misspellings in the formulas in the abstract on this webpage. Read the abstract in the full-text document for correct spelling in formulas, see the downloadable file.

Available from: 2013-06-25 Created: 2013-06-25 Last updated: 2013-07-02Bibliographically approved

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Chotai, Jayanti

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