Probability for Statisticians
Probability for Statisticians is intended as a text for a one year graduate course aimed especially at students in statistics. The choice of examples illustrates this intention clearly. The material to be presented in the classroom constitutes a bit more
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stics are introduced in Section 9.2, while useful linear algebra and the multivariate normal distribution are the subjects of Section 9.3 and Section 9.4. Following the CLT via Stein’s method in Section 11.1, extensions in Section 11.2– 11.3, and application of these CLTs to the bootstrap in Sections 11.4–11.5, there is a large collection of statistical examples in Section 11.6. During presentation of the CLT via chfs in Chapter 14, statistical examples appear in Sections 14.1, 14.2, and 14.4. Statistical applications based on the empirical df appear in Sections 12.10 and 12.12. The highly statistical optional Chapters 16 and 17 were discussed briefly above. Also, the conditional probability Sections 8.5 and 8.6 emphasize statistics. Maximum likelihood ideas are presented in Section A.2 of Appendix A. Many useful statistical distributions contain parameters as an argument of the gamma function. For this reason, the gamma and digamma functions are first developed in Section A.1. Section A.3 develops cumulants, Fisher information, and other useful facts for a number of these distributions. Maximum likelihood proofs are in Section A.4. It is my hope that even those well versed in probability theory will find some new things of interest. I have learned much through my association with David Mason, and I would like to acknowledge that here. Especially (in the context of this text), Theorem 12.4.3 is a beautiful improvement on Theorem 12.4.2, in that it still has the potential for necessary and sufficient results. I really admire the work of Mason and his colleagues. It was while working with David that some of my present interests developed. In particular, a useful companion to Theorem 12.4.3 is knowledge of quantile functions. Sections 7.6–7.11 present what I have compiled and produced on that topic while working on various applications, partially with David. Jon Wellner has taught from several versions of this text. In particular, he typed an earlier version and thus gave me a major critical boost. That head start is what turned my thoughts to writing a text for publication. Sections 8.6, 19.2, and the Hoffman–Jorgensen inequalities came from him. He has also formulated a number of exercises, suggested various improvements, offered good suggestions and references regarding predictable processes, and pointed out some difficulties. My thanks to Jon for all of these contributions. (Obviously, whatever problems may remain lie with me.) My thanks go to John Kimmel for his interest in this text, and for his help and guidance through the various steps and decisions. Thanks also to Lesley Poliner, David Kramer, and the rest at Springer-Verlag. It was a very pleasant experience. This is intended as a textbook, not as a research manuscript. Accordingly, the main body is lightly referenced. There is a section at the end that contains some discussion of the literature.
Contents Use of this Text xiii Definition of Symbols
xviii
Chapter 1. Measures 1. Basic Properties of Measures 1 2. Construction and Extension of Measures 3. Lebesgue–Stie
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