## Mathematical theory of statistics. Statistical experiments and asymptotic decision theory.(English)Zbl 0594.62017

De Gruyter Studies in Mathematics, 7. Berlin - New York: de Gruyter. XII, 492 p. DM 158.00 (1985).
The book deals with the asymptotic justification of statistical methods particularly testing and estimation from decision theoretic viewpoint. As remarked in the introduction this book attempts to provide a ”missing link” between introductory texts like Lehmann, Zacks, Ferguson, Schmetterer and the general thesis advocated by Wald-Le Cam theory, that if the probability models are smooth enough and the sample sizes are large then the statistical analysis of such models can be approximated by the analysis for the normal models. A very interesting feature of the book is the introductory remark at the beginning of each chapter which indicates the historical as well as conceptual development of the material presented in that chapter.
The basic idea is that of testing and the concept of binary experiment for which Neyman-Pearson lemma provides an optimal procedure for a variety of optimality criteria. Using the two error probabilities an order relation between two binary experiments is defined and through this, the concept of sufficiency is introduced. The order relation used is $$''E_ 1$$ is more informative than $$E_ 2''$$ but this ”information” is quite different from other measures of information, such as Fisher, Kullback-Leibler which are commonly used in large parts of statistical theory. However the first six chapters develop the theory of hypotheses testing in a nice way, covering the testing problems in exponential class of densities and the Gaussian distributions.
The author points out that as opposed to the testing problem where there appears to be almost universal agreement on Neyman-Pearson approach, the same is not true in estimation, although some would like to emphasize the method of maximum likelihood as a universal method of construction. But as is well known even in exponential class of densities MLEs may be inadmissible w.r.t. squared error loss. The author here introduces median and mean unbiasedness w.r.t. fairly general loss functions and develops a theory of optimal unbiased estimation. Chapter 8 introduces the general decision theory for experiments and Chapter 9 develops a theory of comparison of experiments in a more general way than was done in Chapter 3. Using the machinery developed so far Chapter 10 discusses asymptotic decision theory in a very general setting. In order to accommodate the (asymptotic) non-parametric analysis, Gaussian distributions in infinite dimensional spaces are studied in Chapter 11. Chapter 12 develops the theory of (stochastic) asymptotic expansions of the likelihood ratios and differentiable experiments. Chapter 13 deals with the asymptotic normality and approximate sufficiency, with applications to testing and estimation. An appendix explains notation and terminology and there is a large up-to-date bibliography. Author, subject and symbol index is also provided.
This is a very well written, tough (at least to the reviewer) mathematical text and would be extremely useful to the research workers interested in asymptotics in general. While the Wald-Le Cam thesis is presented in a way to make it appealing and justifiable, its usefulness is somewhat limited by the fact that the author does not emphasize the rates of convergence. These problems are undoubtedly very difficult and they are model (experiment) specific, but eventually if this theory is to be applied to actual problems the rates of convergence would have to be studied also. One must have some idea as to how large n should be so that the substitution by Gaussian analysis is acceptable to a desired level of accuracy.
The author must be congratulated on a job well done and it is hoped that this text may attract many more able statisticians to probe deeper into these problems.
Reviewer: B.K.Kale

### MSC:

 62F03 Parametric hypothesis testing 62-02 Research exposition (monographs, survey articles) pertaining to statistics 62F10 Point estimation 62C99 Statistical decision theory 62F05 Asymptotic properties of parametric tests 62F12 Asymptotic properties of parametric estimators
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