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Summary In the context of the data and issues discussed by Goldstein and Spiegelhalter, we suggest refinements which can be used by decision makers when confronted with ranking problems associated with ‘league tables’. Two ranking criteria are defined and their performance illustrated for one of the studies reported by Goldstein and Spiegelhalter.

Identification, ranking and selecting hazardous traffic accident locations from a group under consideration is a fundamental goal for traffic safety researchers. Few methods exist that can quantitatively, accurately and easily discriminate between sites that commonly have small and variable observation count periods. One method that embodies all these advantages is the hierarchical Bayesian model, the method proposed in this paper. The particular hierarchical Bayesian approach that we use incorporates expert knowledge about accident sites as a group believed a priori to be exchangeable, the Poisson assumption and a conjugate gamma prior. We then propose three natural strategies for ranking and selecting the most hazardous subgroup of accident locations. Also presented is an especially useful procedure that gives the probability that each particular site is worst and by how much it is worst. All proposed strategies are illustrated using previously published fatality accident data from 35 sites in Auckland, New Zealand.

The high reliability of modern components requires accelerated testing to compare or predict survival or failure rates in the use condition. If the testing of highly reliable components is done in the use condition, the times to failure are too long to observe. Therefore, it is often required to compare or predict mean times to failure in the use condition when the only available data are from a highly accelerated condition. Comparison of failure rates in this situation is possible using frequentist methods but estimation of the individual failure rates is not. Using Bayes methods, both comparison and prediction results are easily possible and computable. The accelerated model in this paper is similar to the model used in health-related research when the data are from a paired experiment. This use of the model to compare and predict survival rates using paired data is also handled easily by the Bayes approach. Let the two failure rates be /spl lambda//sub 1/, /spl lambda//sub 2/; the results are presented as the posterior probability /spl Pscr/r{/spl lambda//sub 1/

These rules for terminating testing involve achieving specified levels of credibility in both: (1) the probability of the posterior estimate of the exponential scale-parameter, /spl theta/, after some, m, of the units have been tested, and (2) the mean of the probability of /spl theta/ over the remaining units, when viewed pessimistically. Sample decision tables and a numerical example illustrate both the sequential and batch testing cases. Large savings in test times can be achieved whenever the first m units present strong evidence in favor of either hypothesis (H/sub 0/: /spl theta/=/spl thetasub 0/ vs. H/sub 1/: /spl theta/=/spl thetasub 1/, /spl thetasub 1/ >

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.

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.

We consider modified empirical Bayes problems in which the prior distribution of $\Theta$ at stage $n + 1$ is $G^{(n+1)}(\theta)$. The Bayes optimality criterion is now given by the sequence of functionals $R(G^{(n+1)}$. The observations $X_1, \cdots, X_n$ are no longer i.i.d so decision procedures are constructed based on modified empirical density estimates for $f_G^{(n+1)}(x)$. Asymptotic optimality together with asymptotic convergence rates is established for two action and estimation problems when the observations are drawn from a member of the one-parameter exponential family.

The purpose of this paper is to obtain weaker sufficient conditions than those given by Robbins [6] for the asymptotic optimality of empirical Bayes procedures. In so doing a general method for providing asymptotic optimality is given which because of its simplicity should prove quite useful not only for the empirical Bayes problem but for other related asymptotic problems as well. Three areas of application are indicated herein.

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