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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.

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.

The use of the maximum entropy principle for determining prior distribution is compared with other techniques in statistical-decision theory for estimating reliability. The comparison is made in the context of estimating the parameter representing the probability of success in a binomial model and the parameter representing the mean time to failure in a simple exponential model. The existence of partial prior information of varying degrees is also assumed. When a quadratic loss function is used, the techniques based on the maximum entropy principle lose much of their appeal.

SUMMARY It is shown that, for quadratic loss and members of the exponential family with an unbiased sufficient statistic having a variance which is quadratic in the parameter, a C2-minimax estimator exists. In addition, the estimator is of simple form and appears to satisfy intuition. Also, it is shown that the 62-minimax estimator is admissible in the usual sense, is 92-admissible and makes the unbiased sufficient statistic inadmissible in S2. Finally, in this situation, the natural conjugate prior distribution is always a member of the Pearson family and is always least favourable. Asymptotic 92-minimax estimators are discussed briefly.

(1969). Joint Continuity of Monotonic Functions. The American Mathematical Monthly: Vol. 76, No. 1, pp. 74-76.