Optimal Sequentially Planned Decision Procedures

Lecture Notes in Statistics Band 79

Norbert Schmitz

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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.


Einband Taschenbuch
Seitenzahl 207
Erscheinungsdatum 28.10.1992
Sprache Englisch
ISBN 978-0-387-97908-3
Verlag Springer Us
Maße (L/B/H) 23.5/15.5/1.2 cm
Gewicht 353 g

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  • 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.