Showing papers in "Test in 1994"

An overview of robust Bayesian analysis

Abstract: Robust Bayesian analysis is the study of the sensitivity of Bayesian answers to uncertain inputs. This paper seeks to provide an overview of the subject, one that is accessible to statisticians outside the field. Recent developments in the area are also reviewed, though with very uneven emphasis.

Characterization of the Jorgensen set in generalized linear models

Abstract: Shanbhag’s clever method for finding the Jorgensen set of the family of Wishart distributions on symmetric matrices is extended here to Wishart distributions on symmetric cones, such as Hermitian matrices on complex numbers or quaternions. The idea is also extended to various other multivariate distributions, including the natural exponential family associated with the set of normal distributions onR with unknown mean and variance.

A predictivistic interpretation of the multivariatet distribution

Abstract: De Finetti type theorems characterize models in terms of invariance. The idea is to take observables, postulate symmetry and then represent the model as a mixture of standard parametric models. If additional conditions are specified, then the mixing measure can be determined. Invariance under the action of special groups of orthogonal transformations may give results on mixtures of parametric normal distributions (Diaconis, Eaton and Lauritzen, 1992). The additional conditions required to determine the mixing measure in this case can be obtained using results in Diaconis and Ylvisaker (1979, 1985). From these results, we obtain a predictivistic characterization of the multivariatet distribution. Furthermore, we state conditions under which then-dimensional law of sequences of random variables is a location mixture of multivariatet distributions. The results are extended to the case of sequences of orthogonally invariant random vectors.

Exponential families with variance functions in\(\sqrt {\Delta P} (\sqrt \Delta )\): Seshadri’s class: Seshadri’s class

Abstract: This paper presents a classification of the setS 3 of all natural exponential families (NEF) on ℝ which have a variance function of the form $$\sqrt {\Delta P} (\sqrt \Delta )$$ , whereP is a polynomial of degree 3 and Δ is an affine function of the mean of the NEF. Particular cases have been considered previosly by V. Seshadri and can be obtained by a Lindsay transform of the NEF with cubic variance, as classified by Marianne Mora.S 3 may be split into six types and we provide a probabilistic interpretation of each of them; in particular, we show that the literature on random mappings provides several examples of discrete elements ofS 3. The final result gives the closure ofS 3 under the topology of weak convergence.

The theory of search from a statistical viewpoint

Abstract: The statistical approach to search is a subject which interfaces with many other fields. This work continues a series of papers which studies these relationship with special emphasis on geometric problems in non-sequential search. A targetT is sought using a series of testsX 1,X 2.... From each test there is an observationY i=f(X i , θ), where θ is an unknown parameter andT=T(θ). Spacing theories and the theories of Voronoi regions can be used to represent the solutions to Bayes and entropy-based formulations to the search problem by partitioning the search field intoconsistent regions where the target is known to lie, given the observations. Other examples are from integration where the target is a function. Covering theory is studied and the last section gives a brief comparison with a sequential approach.

Group-Bayes estimation of the exponential mean: A preposterior analysis

Abstract: In their contribution to the Wald retrospective session at the Fifth Purdue Symposium on Statistical Decision Theory and Related Topics, van Eeden and Zidek (1994) discussed the impact of Wald’s decision theory on the development of statistical science. They approached their assessment from the perspective of a current problem, that of finding a group-Bayes estimator of the exponential mean; they presented several new results in this area. The present paper develops the technical theory underlying the preposterior analysis of van Eeden and Zidek (1994) and it offers some new results as well.

Distance weighted losses for testing and confidence set evaluation

Abstract: The Neyman-Pearson evaluation of testing procedure is often criticized as yielding a decision that is too crude with respect to the decision space and the loss function. Here, we propose some extensions which take into account the notion of distance from the boundary of the confidence set or between the hypotheses. This allows for new evaluation tools, as well as extensions to cases where improper priors could not be used previously. We also reconsider the testing setup as a whole and incorporate new losses which take into account the model choice aspect of testing.

Motivation for the use of discrete distributions in quality assurance

Abstract: By considering a finite population ofN items andS defects, and observing the way defects should be distributed among the items we provide an interesting motivation to the bionomial, negative binomial (geometric) and Poisson distributions for the number of defects in a sample from a production line. The idea is to find out, from physical considerations about the production process, which configurations of defects in items are equally likely. A uniform distribution is assessed on the space generated by these configurations. Then, a distribution for a finite, and subsequently for an infinite population of items is derived.

Asymptotic normality under transformations: a result with Bayesian applications

Abstract: A generalization is provided of the conditions to be satisfied in order to ensure asymptotic normality under transformations. From a Bayesian viewpoint, this result may be used to select an appropriate parameterization or to avoid additional calculations when the parameter of interest does not coincide with the usual parameter of the model.

Operational parameters in Bayesian models

Abstract: Operational parameters are parameters that are defined in terms of the data that are being modelled. This paper shows under what conditions they can be used in place of the usual formal parameters and discusses the advantages of doing this. Example applications include the normal, exponential and uniform model, the multivariate-normal and other multivariate models, finitepopulation versions, and also several new models. Also presented is their relationship to de Finetti’s work on exchangeability and other symmetrybased approaches.

Robust Bayesian analysis given priors on partition sets

Abstract: In a Bayesian analysis, suppose that probability measures may be specified over the subsets partitioning the parameter space Ω, in such a way that they can be combined to form a unique prior measure, defined over all Ω, according to some weights. Should the weights be uncertain, then the class Г of all the probability measures compatible with such uncertainty is specified instead. Situations in which such a class Г is justified are presented and, in these cases, established and quite recent techniques in the field of robust Bayesian analysis are applied to Г. Bounds on posterior expectations are computed, as the prior measure varies in Г, whilst concentration functions and coefficients of divergence are considered when interest lies with comparing functional forms of measures in Г.

An automatic and proper Bayesian estimation analysis of 2×2 contingency tables with one and two fixed margins

Abstract: The Bayesian analysis of a 2×2 contigency table with one or two fixed margins is presented as an estimation problem when using Exponential Family likelihoods with two or one free parameters, respectively. The computation of the Jeffreys priors for one or two fixed marginals is then straightforward for the canonical parameter. Jeffreys priors are proper distributions, despite of the fact that the parameter spaces are unbounded. Coupling the Jeffreys prior with exponential likelihoods thus yields a proper and automatic Bayesian analysis. Two contingency tables with one and two fixed margins are then analysed.

Bayesian estimation of a normal mean parameter using the Linex loss function and robustness considerations

Abstract: This paper presents a derivation of an explicit analytical form for the Bayes estimator of the normal location parameter using the Linex loss function with a general class of prior distributions. Exact and approximate results based on Pericchi and Smith’s paper (1992) are given, where the priors are double-exponential and Studentt, respectively. The results of this paper provide a link between the robust Bayesian analysis for the normal location parameter when adopting either the Linex loss function or the squared error loss function.

The central limit theorem for empirical processes on V-Č classes: A majorizing measure approach

Abstract: Alexander (1987) gave necessary and sufficient conditions for the central limit theorem for empirical processes on Vapnik-Cervonenkis classes of functions. In this paper we present an alternative version of his result using Talagrand’s analytic characterization of pregaussianness (the majorizing measure condition). Our proof can be directly extended to give the corresponding result in the non-gaussian stable case.

Prediction based response surface data obtained with random blocking

Abstract: In many experiments, a main purpose is prediction of a future response, future to the data obtained from the experiment. Frequently, such experiments have to be run in blocks where the block effects are random. In this paper, we describe a Bayesian approach to the problem of prediction, given data which have been obtained from a response surface design with random blocking. The predictive distribution involves a posterior distribution which is not available in closed form, and we outline a Gibbs sampling procedure to carry out the determination of the needed predictive distribution.

Bayesian robustness of the empirical distribution

Abstract: A new method for dealing with possibly contaminated data, when there exists prior information about the contamination procedure, is proposed and studied in order to obtain robust estimators. A class of Bayesian robust estimators is then established and its properties studied. Applications of the proposed method are given.

The use of auxiliary information for solving non-response problems

Abstract: The unavailability of information from some sampled units complicates the use of current sampling methods. The classical solution is to select a subsample from within the non-respondents. A random non-response mechanism provides an alternative model within the randomization approach. A superpopulation point of view suggests use of a predictor. Shrinkage techniques are used to derive methods for dealing with non-responses. The accuracy of the analysed models is compared using Monte Carlo experiments.

Bayesian prediction for business mortality analysis

Abstract: This paper is concerned with business mortality analysis in a Bayesian setting. We assume that a businessman startsN businesses at different points of time and at a certain epoch referred to as the ‘present’, the failure times of the failed businesses and the survival times of the still surviving businesses are recorded. Bayesian prediction for business mortality or survival up to a future time pointt0 is made under the assumptions of a Weibull business survival distribution (WBSD) and the squared error loss function (SELF). The results are extended to the situation where one of the observed times of business failure may possibly be an outlier.