Klaus J. Miescke

Bayesian look ahead one-stage sampling allocations for selection of the best population

From k independent normal populations with unknown means and a common known variance, Bayesian selection procedures are considered for finding that population which has the largest mean. Suppose that a first stage has been completed already, where k samples have been observed, which may be of different sizes. Let there be m additional observations allowed to be taken at a future second stage. The problem of interest treated here is how to allocate these m observations in an optimum way among the k populations, given all of the information, prior and first stage observations, gathered so far. This allocation problem will be formulated and discussed in a more general framework, and specific results will be presented for the normal case with independent conjugate priors under linear loss.

Bayesian look ahead one stage sampling allocations for selecting the largest normal mean

From two independent normal populations with unknown means and a common known variance, samples of unequal sizes are observed at stage 1. The goal is to find that population with the larger mean. Using the Bayes approach, optimum allocations ofm additional observations, at stage 2, are derived under the linear and the 0–1 loss.

Weibull models for the statistical analysis of dental composite data: aged in physiologic media and cyclic-fatigued

The modulus of rupture or flexural strength of dental composites aged in distilled water and saline solution or cyclic-fatigued to fracture in distilled water was analyzed by use of Weibull statistics. Two- and three-parameter Weibull models were applied to the data. For the case of the dental composites aged for five distinct time periods, the most appropriate model was a two-parameter Weibull model for each separate aging time. For the cyclic-fatigue data, a three-parameter accelerated failure time model with a Weibull baseline distribution and scale parameter, depending on the variable cycle, was appropriate.

On selecting the largest success probability under unequal sample sizes

Abstract Let π1,…,πk be k≥3 independent binomial populations, from which Xi∼B(ni,Pi), i = 1,…,k, respectively, have been observed. The problem under concern is to find that population which is associated with the largest of the unknown ‘success probabilities’ p1,…,pk. Under the ‘0–1’ loss, some linear loss which occurs in gambling, and a general monotone, permutation invariant loss, interesting properties of Bayes rules are studied for priors which are permutation invariant, as well as for priors which are not invariant but have a (DT)-posterior density with respect to some symmetric measure. Examples of independent beta-priors are included.

Simultaneous selection and estimation in general linear models

The problem of selecting the largest treatment parameter, and simultaneously estimating the selected treatment parameter, in a general linear model is considered in the decision theoretic Bayes approach. Both cases, where the error variance is known or unknown, are included. Bayes decision rules are derived for noninformative priors and for normal priors. The problem of finding Bayes designs, i.e. designs that have minimum Bayes risk, within a given class of designs is also discussed.

A Bayesian decision theoretic approach to directional multiple hypotheses problems

A multiple hypothesis problem with directional alternatives is considered in a decision theoretic framework. Skewness in the alternatives is considered, and it is shown that this skewness permits the Bayes rules to possess certain advantages when one direction of the alternatives is more important or more probable than the other direction. Bayes rules subject to constraints on certain directional false discovery rates are obtained, and their performances are compared with a traditional FDR rule through simulation. We also analyzed a gene expression data using our methodology, and compare the results to that of a FDR method.

On the performance of subset selection rules under normality

From k normal populations N(θ1,σ12),…,N(θk,σk2), where the means θ1,…,θk∈R are unknown, and the variances σ12,…,σk2>0 are known, independent random samples of sizes n1,…,nk, respectively, are drawn. Based on these observations, a non-empty subset of these k populations of preferably small size has to be selected, which contains the population with the largest mean with a probability of at least P∗ at every parameter configuration. Several subset selection rules which have been proposed in the literature are compared with Bayes selection rules for normal priors under two natural type of loss functions. Two new subset selection rules are considered.

Optimum Two-Stage Selection Procedures for Weibull Populations.

Abstract Let π 1 ,…, π k be Weibull populations with a common known shape parameter, and with unknown scale parameters. The goal is to find the population with the largest scale parameter. From each population, Type II-censored observations are available at two stages, where censoring at stage 1 (2) occurs at the q -th ( r -th) failure. Two-stage procedures with screening at the first stage are considered which are optimum permutation invariant in terms of the risk with respect to a large class of loss functions. For the procedure with a fixed subset size at stage 1, the least favorable parameter configuration under the indifference zone approach is of the slip-page type, which makes it feasible to control the infimum of the probability of a correct selection. Some extensions of the results are discussed at the end.

Minimax multiple t-tests for comparing k normal populations with a control☆

Abstract Let π 1 ,·, π k be k normal populations with unknown means θ 1 ,·, θ k , and a common unknown variance σ 2 > 0. Based on independent samples of sizes n 1 ,·, n k , the populations are to be partitioned into two sets, where the first one contains all π i with θ i ≥ θ 0 , and where the other one contains the rest. At first it is assumed that θ 0 is known. Under an additive ‘ a i − b i ’ loss function a minimax procedure is derived which is of a simple natural form. The proof of minimaxity makes use of the Bayes approach and involves a sequence of nonsymmetric priors, which play a similar role as a least favorable prior in simpler problems. Analogous results are presented for the case that θ 0 is not known. In this case, a control normal population is assumed to exist from which an additional sample of size n 0 can be drawn.

A stochastic model for paired comparisons of social stimuli

Abstract A stochastic model for paired comparisons of multiattribute social stimuli is proposed where one objective is to find the relative importance of the attributes for a judge. The model can be conceived as a special strict binary utility model, i.e., a BTL-model, and is related to of the stimuli are linear combinations of functions of the attributes of the stimuli. The model neither assumes that the functions are fixed in advance nor that different judges have the same set of functions. The choice among such functions, however, is admitted only within a finite scope. Within the framework of exponential families, maximum likelihood estimators and tests are derived and applied to data coming from two psychological experiments.