First and Second Order Efficiency of Sequential Designs in a Nonlinear Situation with Applications
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Abstract
In many statistical problems, optimal designs may be infeasible with their dependence on the parameters, which are unknown a priori. In such circumstances, choosing the next design points adaptively by making good use of information being collected so far, namely, designing the experiment sequentially, is appropriate. In this study, the problem of estimating a nonlinear function of the parameters, in particular, the product of finitely many parameters, is investigated and generalized under the umbrella of one-parameter exponential family. A Bayesian formulation is adopted with assumptions that the parameter are independent a priori and have conjugate prior distributions. The difficulties involved in computing explicit Bayes solutions lead to the derivation of asymptotic first- and second-order lower bound. Sequential sampling procedures of first-order efficiency are characterized by the development of necessary and sufficient conditions in a more general sense. A three-stage sequential sampling scheme is shown to be asymptotic efficient by satisfying the necessary and sufficient conditions and later adapted to two application problems, which further confirms the theoretical results through Monte Carlo simulations.
Table of Contents
Introduction -- Preliminary -- First-order asymptotic efficiency of sequential designs for estimating product of K means in the exponential family case -- Applications to reliability estimation and risk assessment -- Second-order asymptotic efficiency in sequential designs for estimating product of means in the exponential family case -- Summary ad conclusion -- Appendix A. Figures -- Appendix B. Tables
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Ph.D.