Optimal sequentially planned decision procedures
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Bibliographic Information
Optimal sequentially planned decision procedures
（Lecture notes in statistics, 79）
Springer-Verlag, c1993
- : us
- : gw
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Note
Includes bibliographical references (p. [199]-205) and index
Description and Table of Contents
- Volume
-
: us ISBN 9780387979083
Description
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 conse quence, the SPRT's played a dominating role in the further development of sequential analysis and, more generally, in theoretical statistics.
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 of policies 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.
- Volume
-
: gw ISBN 9783540979081
Description
This volume is concerned with statistical procedures where the data are collected in sequentially designed groups. The basic premise here is that the expected total sample size is not always the appropriate criterion for evaluating statistical procedures, especially for nonlinear sampling costs (eg. additive fixed costs) and in clinical trials. In fact, this criterion seems to have been a hindrance to the practical use of Wald's sequential probability ratio test (SPRT) despite its well-known optimum properties. This volume systematically develops decision procedures which retain the possibility of early stopping and remove some of the disadvantages of one-at-a-time sampling. In particular, for generalizations of the SPRT algorithms, methods for computing characteristics (such as operating characteristics or power functions, expected sampling costs, etc) are developed and implemented. The procedures turn out to be optimal in a Bayesian sense as well as for problems with side conditions (eg. specified bounds on error probabilities or expected sampling costs). A theory of optimal sampling is developed in order to prove the various properties of the procedures.
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