A Single-Sample Multiple-Decision Procedure for Selecting the Multinomial Event Which Has the Highest Probability

Bechhofer, Robert E.; Elmaghraby, Salah E.; Morse, Norman

doi:10.1214/aoms/1177706362

Cited by 123 publications

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“…The approach which is essentially that of [10], is most simply given in terms of the general finite decision problem discussed in Section 1. In such a problem, which is assumed to remain invariant under a finite group G, suppose an ordering ~ is introduced among the procedures (read 'P ~ 'P 1 as 'P 1 is at least as good as fiJ) • LEMMA 2~ (a) If the ordering is such that Lemma 2, in conjunction with the Bahadur-Goodman theorem proves, for example, that the procedure rp<o> of that theorem maximizes the minimum probability of a correct decision whenever the best population is sufficiently much better than the second best (and hence establishes an optimum property of the procedures given in [4], [5], [6] and [11]). This result was proved by the method of least favorable distributions by Hall (1959).…”

Section: E D •• = D;)mentioning

confidence: 99%

On a Theorem of Bahadur and Goodman

Lehmann

2011

Selected Works of E. L. Lehmann

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1. Introduction. This note is concerned with the problem of selecting the best one (or any other specified number) of several populations. It is restricted to the symmetric case where typically the observations consist of samples of equal size from the different populations. For certain families of distributions, and Bahadur and Goodman {1952) have proved that the natural selection procedure uniformly minimizes the risk among all symmetric procedures for a large class of loss functions. In Section 2 we give an alternative proof of this theorem, and in Section 3 show that the theorem implies many other optimum properties including one obtained in a different manner by Hall (1959).The problem of selecting the best one of s populations is a finite decision prob- for all x. We suppose that the distribution P6 of X depends on the parameter() and that the loss resulting from decision d, when () is the true parameter value isCorresponding to the symmetry assumed for the selection problem, we shall assume that the problem is invariant under the finite transformation group G = {g1, · · · , gN}: if the distribution of X is d[X} = P6, the random variable g;X has distribution d [g,X] = Pu,6 where g, and g; are 1:1 mappings respectively of the sample space and of the parameter space onto themselves; furthermore there exist transformations g1*, · · · , gN * of the decision space (i.e. permutations of d1 , · · · , d.) such that for any i, j and ()A procedure IP is then said to be invariant if (2) g*tp(x) = IP(gx) for all x and g.The procedure taking on the value tp(gx) at the point x will be denoted by tpg, and (2) can then be written as g* tpg-1 = IP· To prove that their procedure uniformly minimizes the risk among all invariant procedures, Bahadur and Goodman first characterize the totality of invariant procedures. An altenative proof can be based on the following lemma concerning general finite invariant decision problems.

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Section: E D •• = D;)mentioning

confidence: 99%

On a Theorem of Bahadur and Goodman

Lehmann

2011

Selected Works of E. L. Lehmann

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“…5. Bechofer et al ( 1959) have suggested the ratio pk1 /pk2 for a similar problem. For two reasons, this seems less appropriate than t>.…”

Section: Notesmentioning

confidence: 99%

Interpreting Modal Frequencies To Measure Social Norms

Jacobsen

Voordt

1980

Sociological Methods & Research

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“…The set of problem configurations satisfying this constraint on the difference in means is called the preference zone. The paper Bechhofer (1954) is considered the seminal work, and early work is presented in the monograph Bechhofer, Kiefer, and Sobel (1968). Some compilations of the theory developed in the area can be found in R. E. Bechhofer (1995), Swisher, Jacobson, and Yücesan (2003), Kim and Nelson (2006) and Kim and Nelson (2007).…”

Section: Introductionmentioning

confidence: 99%

“…The first IZ procedures presented in Bechhofer (1954), Paulson (1964), Fabian (1974), Rinott (1978), Hartmann (1988), Hartmann (1991), Paulson (1994) satisfy the IZ guarantee, but they usually take too many samples when there are many alternatives, in part because they are conservative: their probability of correct selection (PCS) is much larger than the probability specified by the user (Wang and Kim 2013). One reason for this is that these procedures use Bonferroni's inequality, which leads then to sample more than necessary.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic validity of the Bayes-inspired Indifference Zone procedure: The non-normal known variance case

Toscano-Palmerin

Frazier

2015

2015 Winter Simulation Conference (WSC)

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We consider the indifference-zone (IZ) formulation of the ranking and selection problem in which the goal is to choose an alternative with the largest mean with guaranteed probability, as long as the difference between this mean and the second largest exceeds a threshold. Conservatism leads classical IZ procedures to take too many samples in problems with many alternatives. The Bayes-inspired Indifference Zone (BIZ) procedure, proposed in Frazier (2014), is less conservative than previous procedures, but its proof of validity requires strong assumptions, specifically that samples are normal, and variances are known with an integer multiple structure. In this paper, we show asymptotic validity of a slight modification of the original BIZ procedure as the difference between the best alternative and the second best goes to zero, when the variances are known and finite, and samples are independent and identically distributed, but not necessarily normal.

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A Single-Sample Multiple-Decision Procedure for Selecting the Multinomial Event Which Has the Highest Probability

Cited by 123 publications

References 0 publications

On a Theorem of Bahadur and Goodman

On a Theorem of Bahadur and Goodman

Interpreting Modal Frequencies To Measure Social Norms

Asymptotic validity of the Bayes-inspired Indifference Zone procedure: The non-normal known variance case

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