Inference and optimal design in Bayes and classical problems

Saurabh Mukhopadhyay, Purdue University

Abstract

The emphasis in this work is on derivation of optimal Bayes inferences and designs in relatively unexplored models or formulations. Linear and non-linear regression models are considered for finding Bayes as well as classical optimal designs. The work focuses principally on the three following families of problems: Part I: Inference. In this work, we consider inference problems for location parameters. The idea is that if one can produce priors for which the posterior densities are uniformly close to the likelihood function, then the corresponding Bayesian inference should also be close to classical inference, at least for location parameters. We describe a large family of prior distributions meeting this goal. Apart from obtaining approximations for the posterior density itself, we also derive uniform approximations to the Bayes rule and the posterior expected loss. We also demonstrate that for these priors, the sampling distributions of the Bayes rule and the classical unbiased estimate are close uniformly in the parameter and that all $100(1 - \alpha)$% Bayesian HPD sets have a classical coverage probabilities uniformly close to $1 - \alpha$ as well. Part II: Design. (i) Compromise designs. Multiple linear regression models are considerd with emphasis on construction of designs that provide a guaranteed prespecified efficiency simultaneously for each of a collection of different statistical problems. It is usually always the case that the experimenter is simultaneously interested in more than one inference problem using the same data and the sheer difference in the nature of the different problems makes combining them in terms of a single loss function undesirable. On the other hand, designs optimal in one problem may be inefficient in other similar problems and yet the statistician has to select one single design. Various heteroscedastic models are studied and the design problem described above is addressed and solved. These problems are addressed in Bayesian as well as classical frameworks and the solutions are compared. (ii) Nonlinear regression models and construction of Bayesian optimal designs. We consider exponential growth models which are of great practical use for describing growth of organisms over time. We have successfully used Tchebycheff-system and convexity arguments in some cases. We consider D-optimum designs with or without prior information and give some qualitative understanding of the nature of the optimal design depending on the situation. In particular, we try to draw parallels with the linear case. (Abstract shortened by UMI.)

Degree

Ph.D.

Advisors

DasGupta, Purdue University.

Subject Area

Statistics

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