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Probability for statisticians. (English) Zbl 0951.62005

Springer Texts in Statistics. New York, NY: Springer. xvii, 585 p. DM 159.00; öS 1161.00; sFr 144.00; £55.00; $ 79.95 (2000).
This text covers a broad range of probability theory with special emphasis on fields which are useful for statistics. To point out some of them: Included are discussions of the quantile transformation, bootstrapping, Winsorization, Edgeworth expansions, \(L\)-statistics, linear rank statistics, and finite sampling. The table of contents is as follows: Measures; Measurable Functions and Convergence; Integration; Derivatives via signed Measures; Measures and Processes on Products; General Topology and Hilbert Spaces; Distribution and Quantile Functions; Independence and Conditional Distributions; Special Distributions; WLLN; SLLN; LIL; and Series; Convergence in Distribution; Brownian Motion and Empirical Processes; Characteristic Functions (chfs); CLTs via chfs; Infinitely Divisible and Stable Distributions; Asymptotics via Empirical Processes; Asymptotics via Stein’s Approach; Martingales; Convergence in Law on Metric Spaces.
The appendix gives a ‘Distribution Summary’ with explanation to the Gamma function, Maximum Likelihood and examples of statistical models.
One characteristic unique to this text is the presentation of different approaches to the CLT. It is stated first via Stein’s method in the chapter on convergence in distribution. CLTs via chfs follow, and Hoeffding’s combinatorial CLT is dealt with in ‘Asymptotics via Stein’s Approach’. The presentation is in the style: definition – proposition/theorem – proof. More detailed motivations are concentrated in special paragraphs. Questions are included to open the view for further generalizations and motivate the reader to think themselves. Exercises are everywhere.

MSC:

62-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics
60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory
60F05 Central limit and other weak theorems
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