Sequential analysis. Tests and confidence intervals.

*(English)*Zbl 0573.62071
Springer Series in Statistics. New York etc.: Springer-Verlag. XI, 272 p. DM 108.00 (1985).

Sequential analysis is one of the very few theories which started nearly immediately with a monograph - namely with the fundamental book ”Sequential analysis” of A. Wald (1947). Despite the unquestionable advantage that one obtained very early a series of important results this had, on the other hand, the consequence that Wald’s analysis and especially the sequential probability ratio test (SPRT) played a (too) important role. The dominance of this special test can still be perceived, e.g. in the often-used text-book of B. K. Ghosh, Sequential tests of statistical hypotheses. (1970; Zbl 0223.62097).

On the other hand Wald’s theory and his SPRT were nearly ignored by practical statistics. Some reasons for the discrepancy between the satisfying theoretical results and the missing acceptance by practice may be, that the SPRT’s have no upper bound for the number of observations, that the surprising Wald-Wolfowitz-optimality of the SPRT’s applies only to the problem of testing simple hypotheses, that taking observations only one by one is too expensive etc.

Starting with the paper ”A modification of the SPRT to reduce the sample size”, Ann. Math. Stat. 31, 165-197 (1960; Zbl 0089.355) of T. W. Anderson therefore truncated sequential tests - especially tests with nonlinear stopping boundaries - and repeated significance tests were investigated. For a deeper understanding of those tests powerful mathematical methods, as presented e.g. in the important research monograph by M. Woodroofe, Nonlinear renewal theory in sequential analysis. CBMS-NSF Reg. Conf. Ser. Appl. Math. 39 (1982; Zbl 0487.62062), are used.

The main goal of the present book which is written by a very competent author is ”to review these recent developments for the most part in the specific framework of Wald’s book, i.e., sequential hypothesis testing in a non-Bayesian, non-decision-theoretic context”.

The headings of the chapters may give a first impression of the concept and of the contents: I. Introduction and examples; II. The sequential probability ratio test; III. Brownian approximations and truncated tests; IV. Tests with curved stopping boundaries; V. Examples of repeated significance tests; VI. Allocation of treatments; VII. Interval estimation of prescribed accuracy; VIII. Random walk and renewal theory; IX. Nonlinear renewal theory; X. Corrected Brownian approximations; XI. Miscellaneous boundary crossing problems. Appendices: Brownian motion; Queueing and insurance risk theory, Martingales and stochastic integrals; Renewal theory.

This list of contents also indicates that the book is not intended as a complete description of sequential analysis - e.g. the discussion of Bayesian sequential tests is omitted - but more as a research monograph which summarizes recent results especially in connection with repeated significance tests - a field where the author himself has made many important contributions.

The book has two faces: On the first hand the author says that he restricts himself on very simple models, that the formal mathematical prerequisites for reading this book have been held to a minimum and that he tries to minimize difficult probability calculations. Consequently, he often gives informal proofs, records the results of algebraic computations, skips the regularity conditions which allow the developments and gives additional results without proofs.

On the other hand, for a real understanding of the book an advanced knowledge of mathematical statistics and a strong mathematical background seem to be necessary. A plenty of substantial material is covered by this monograph. However, for several proofs it seems to be inevitable to consult the original papers.

Chapters III-V form the core of the book. Here Brownian approximations for truncated sequential tests, triangular tests, tests with curved stopping boundaries and modified repeated significance tests are considered. Besides these asymptotic approximations, for many cases numerical examples are given to illustrate the accuracy of these approximations.

The book is very carefully written (the only misprint worth mentioning the reviewer found on page 242) and excellently organized. It constitutes a very important and stimulating contribution to the field of sequential analysis and it is highly recommended to all those who are working in this field.

On the other hand Wald’s theory and his SPRT were nearly ignored by practical statistics. Some reasons for the discrepancy between the satisfying theoretical results and the missing acceptance by practice may be, that the SPRT’s have no upper bound for the number of observations, that the surprising Wald-Wolfowitz-optimality of the SPRT’s applies only to the problem of testing simple hypotheses, that taking observations only one by one is too expensive etc.

Starting with the paper ”A modification of the SPRT to reduce the sample size”, Ann. Math. Stat. 31, 165-197 (1960; Zbl 0089.355) of T. W. Anderson therefore truncated sequential tests - especially tests with nonlinear stopping boundaries - and repeated significance tests were investigated. For a deeper understanding of those tests powerful mathematical methods, as presented e.g. in the important research monograph by M. Woodroofe, Nonlinear renewal theory in sequential analysis. CBMS-NSF Reg. Conf. Ser. Appl. Math. 39 (1982; Zbl 0487.62062), are used.

The main goal of the present book which is written by a very competent author is ”to review these recent developments for the most part in the specific framework of Wald’s book, i.e., sequential hypothesis testing in a non-Bayesian, non-decision-theoretic context”.

The headings of the chapters may give a first impression of the concept and of the contents: I. Introduction and examples; II. The sequential probability ratio test; III. Brownian approximations and truncated tests; IV. Tests with curved stopping boundaries; V. Examples of repeated significance tests; VI. Allocation of treatments; VII. Interval estimation of prescribed accuracy; VIII. Random walk and renewal theory; IX. Nonlinear renewal theory; X. Corrected Brownian approximations; XI. Miscellaneous boundary crossing problems. Appendices: Brownian motion; Queueing and insurance risk theory, Martingales and stochastic integrals; Renewal theory.

This list of contents also indicates that the book is not intended as a complete description of sequential analysis - e.g. the discussion of Bayesian sequential tests is omitted - but more as a research monograph which summarizes recent results especially in connection with repeated significance tests - a field where the author himself has made many important contributions.

The book has two faces: On the first hand the author says that he restricts himself on very simple models, that the formal mathematical prerequisites for reading this book have been held to a minimum and that he tries to minimize difficult probability calculations. Consequently, he often gives informal proofs, records the results of algebraic computations, skips the regularity conditions which allow the developments and gives additional results without proofs.

On the other hand, for a real understanding of the book an advanced knowledge of mathematical statistics and a strong mathematical background seem to be necessary. A plenty of substantial material is covered by this monograph. However, for several proofs it seems to be inevitable to consult the original papers.

Chapters III-V form the core of the book. Here Brownian approximations for truncated sequential tests, triangular tests, tests with curved stopping boundaries and modified repeated significance tests are considered. Besides these asymptotic approximations, for many cases numerical examples are given to illustrate the accuracy of these approximations.

The book is very carefully written (the only misprint worth mentioning the reviewer found on page 242) and excellently organized. It constitutes a very important and stimulating contribution to the field of sequential analysis and it is highly recommended to all those who are working in this field.

Reviewer: N.Schmitz

##### MSC:

62Lxx | Sequential statistical methods |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

60K05 | Renewal theory |

60K15 | Markov renewal processes, semi-Markov processes |