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This article considers the variation of the Secretary Problem where at any stage each of the last m applicants are available for employment. We show that the form of the stopping rule which maximizes the probability of employing the best applicant is: stop at the first stage, from r* onwards, when the relatively best applicant is about to become unavailable. A simple algorithm for evaluating r* and the probability of employing the best applicant is outlined.

A set of unknown normal means (treatment effects, say) {θ1, θ2, …, θk } is to be investigated. Two common questions in analysis of variance and ranking and selection are as follows: (a) What is the strength of evidence against the hypothesis H 0 of equality of means? (b) If H 0 is false, which mean is the largest (or smallest)? A Bayesian approach to the problem is taken, leading to calculation of the posterior probability of H 0 and the posterior probabilities that each mean is the largest, conditional on H 0 being false. A variety of exchangeable, nonexchangeable, informative, and noninformative prior assumptions are considered. Calculations involve, at worst, only low-dimensional numerical integration, in spite of the fact that the dimension k can be arbitrarily large. As an example, Table 1 presents, for each baseball team in the National League in 1984, the highest batting average obtained by any player on the team with at least 150 at bats. The observed batting averages are treated as sampl...

A Bayesian approach is given for various kinds of empirical Bayes problems. In particular it is shown that empirical Bayes procedures are really non-Bayesian, asymptotically optimal, classical procedures for mixtures. In some situations these procedures are Bayes with respect to some prior and in other situations, there is no prior for which they are Bayes. Several examples of these concepts are given as well as a general theory showing the difference between an empirical Bayes model and a Bayes empirical Bayes model.

The purpose of this paper is to make the reader aware of the applicability and advantage of a particular mathematical model. The application is typified by an example and the advantage is via confidence intervals; that is, shorter confidence intervals are possible using the model than if one ignores it, providing the applicability is valid. It is also shown that an improved estimate can be obtained through use of the model. Let f(y|μ,σ) be a normal density with mean μ and variance σ2 and let g(μ|λ, β) be a normal density with mean λ and variance β2. A sequence y 1, y 2, …, y n+1 of independent observations from the mixture of f and g can be considered as follows: An unobservable μi is first drawn from g(μ|λ, β) and then yi which can be observed is drawn from f(y|μi, σ). Confidence intervals on μ n+1 are obtained which are based on the observations y 1, …, y n+1 and which are shorter than the standard interval based on y n+1 only for any n. Shorter intervals are obtained for two cases: (i) λ unkno...

The purpose of this paper is to present an easily applied test useful in determining whether or not a supplier's data have been truncated. The proposed test has the following desirable properties: (i) it is the uniformly most powerful rank test, (ii) it is asymptotically uniformly most powerful, (iii) power computations can easily be made for arbitrary sample sizes, formulas for such computations being given in the paper. Although formulated in the context of verifying a supplier's data, the test can be applied to other situations in which false representation of data in the form of truncation is important. Such is the case, for example, in reliability demonstrations or legal suits involving physical measurements.