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Sequential nonparametrics. Invariance principles and statistical inference. (English) Zbl 0583.62074
Wiley Series in Probability and Mathematical Statistics. New York etc.: John Wiley & Sons, Inc. XV, 421 p. (1981).
A typical objective of the author is to find the asymptotic operating characteristics of sequential procedures based on linear rank statistics. This objective is realized by a long argument including the demonstration that the linear rank statistics can be looked at as a martingale and then using Strassen’s theorem on the embedding of martingales in Brownian motions. A good way to see the basic argument is to work backwards from the objective to the basic. For example, start at pages 240 and 247 for the two-sample problem. (Incidentally, for this problem the \(c_ i\) are more appropriately defined on p. 240 than p. 90, where the problem is introduced.)
There are about 400 references with more than 100 by the author. The work develops in many interwoven directions and the results are too complicated to give here. Some sections, however, are self-contained, such as Section 2.7, which is a summary of results on boundary crossing probabilities of Brownian motions.
The presentation suggests two lines of research. Sequential procedures, particularly those presented by the author, have had limited use in the nearly 40 years since their introduction by Wald. Perhaps the methodologists will find this book a convenient reference in the development of more appealing procedures. Also, as noted above, the analysis and results appear complicated. Future research should unify the area and simplify the results.

MSC:
62L10 Sequential statistical analysis
62-02 Research exposition (monographs, survey articles) pertaining to statistics
62G10 Nonparametric hypothesis testing
60G44 Martingales with continuous parameter