Optimal Sequentially Planned Decision Procedures

by Norbert Schmitz

Overview

Learning from experience, making decisions on the basis of the available information, and proceeding step by step to a desired goal are fundamental behavioural qualities of human beings. Nevertheless, it was not until the early 1940's that such a statistical theory - namely Sequential Analysis - was created, which allows us to investigate this kind of behaviour in a precise manner. A. Wald's famous sequential probability ratio test (SPRT; see example (1.8» turned out to have an enormous influence on the development of this theory. On the one hand, Wald's fundamental monograph "Sequential Analysis" ([Wa]*) is essentially centered around this test. On the other hand, important properties of the SPRT - e.g. Bayes­ optimality, minimax-properties, "uniform" optimality with respect to expected sample sizes - gave rise to the development of a general statistical decision theory. As a consequence, the SPRT's played a dominating role in the further development of sequential analysis and, more generally, in theoretical statistics.

Product Details

ISBN-13:	9780387979083
Publisher:	Springer New York
Publication date:	10/28/1992
Series:	Lecture Notes in Statistics , #79
Pages:	207
Product dimensions:	6.10(w) x 9.25(h) x 0.02(d)

Table of Contents

I. Introduction.- § 1 Sequential statistical procedures.- § 2 Objectives of sequential analysis.- § 3 Historical remarks on the development of sequential analysis.- § 4 Examples of sequential procedures; purely sequential statistical decision procedures.- § 5 Objections to purely sequential statistical decision procedures.- § 6 Sequentially planned statistical procedures.- II. Optimal sequential sampling plans.- § 1 Problems of optimal sampling.- § 2 Optimal sampling plans for finite horizon.- § 3 Existence of optimal sampling plans for general A.- § 4 Optimal sampling plans for the Markov case.- III. Sequentially planned tests; sequentially planned probability ratio tests.- § 1 Notation.- § 2 The iid case.- § 3 Sequentially planned probability ratio tests.- § 4 Algorithms for computing the OC- and ASC-function of SPPRT’s in the iid case.- § 5 Remarks on the implementation of the algorithms; Examples.- § 6 Remarks on the comparison of the methods and on convergence-improvements for the BF-/EV- method.- IV. Bayes-optimal sequentially planned decision procedures.- § 1 Introduction.- § 2 Bayes-procedures.- § 3 A posteriori-distributions.- § 4 Bayes-optimal sampling plans; Markov case.- V. Optimal sequentially planned tests under side conditions.- § 1 Decision problems with side conditions.- § 2 Characterizations of optimal sequentially planned decision procedures.- § 3 Sequentially planned tests for simple hypotheses in the iid case.- § 4 The modified Kiefer-Weiss problem in the iid case.- § 5 Locally optimal sequentially planned tests in the dominated iid case.- § 6 Remarks on the monotonicity of the power functions of SPPRT’s and GSPPRT’s.- Appendix A: Mathematical models for sequentially planned sampling procedures.- § A.1 The concept ofpolicies by Mandelbaum and Vanderbei.- § A.2 The concept of tactics by Krengel and Sucheston.- § A.3 The concept of decision functions by Washburn and Willsky.- § A.4 The concept of stopped decision models by Rieder.- Appendix B: Implementation of the algorithms EV, BF and ILE; Diophantine Approximation.- § B.1 Listing of the modules.- § B.2 Diophantine approximation.- Appendix C: References, Bibliography.