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Mathematical methods of statistics. Reprint of the 1946 original. (English) Zbl 0985.62001
Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press. xvi, 575 p. (1999).
The purpose of this classical textbook from 1946 [Princeton Mathematical series. 9. Princeton N. J.: Princeton University Press (1946; Zbl 0063.01014)] is to join the two major lines of development in the field of mathematical statistics: while British and American statisticians were developing the science of statistical inference, French and Russian probabilists transformed the calculus of probability into a purely mathematical theory. The result of Cramér’s work is a rigorous mathematical exposition of the theory of estimation and the theory of testing statistical hypotheses. The problems are cleverly selected, and the methods of solution are clear and beautiful, and various practical numerical examples are presented; therefore the book is an excellent introduction to mathematical statistics.
The book is not an elementary and easy reading, and a good knowledge of classical calculus is necessary. All the auxiliary mathematical tools beyond standard calculus are given in the first part of the book, especially elements of measure theory. The second part is devoted to probability theory. The third part is basic and deals with mathematical statistics. With a volume of 240 pages only, it contains quite rich material. The exposition is mainly inductive.
In chapters 25–26 the preliminary characteristics of the theory of estimation are given, then certain special problems are considered in chapters 27–31 (characteristics of sampling distributions and their asymptotic properties, goodness-of-fit tests, and tests of significance for parameters), and after that the basic theoretical results of the theory of estimation and testing hypotheses are given in chapters 32–35. More complicated and special problems are treated in chapters 36–37, namely analysis of variance and some regression problems. In these two chapters, the presentation is somewhat schematic and there are no practical examples. Nowadays, analysis of variances and regression analysis have been developed drastically; we mention, e.g., the theory of nonlinear regression and errors-in-variables models.
Harald Cramér was the director of the Institute of Mathematical Statistics at the University of Stockholm.
Contents: First part (chapters 1–12), Mathematical introduction. Chapters 1–3, Sets of points. Chapters 4-7, Theory of measure and integration in \(\mathbb R^1\). Chapters 8–9, Theory of measure and integration in \(\mathbb R^n\). Chapters 10–12, Various questions (including Fourier integrals and matrices).
Second part (chapters 13–24), Random variables and probability distributions. Chapters 13–14, Foundations. Chapters 15–20, Variables and distributions in \(\mathbb R^1\). Chapters 21–24, Variables and distributions in \(\mathbb R^n\).
Third part (chapters 25–37), Statistical inference. Chapters 25–26, Generalities. Chapters 27–29, Sampling distributions. Chapters 30–31, Tests of significance, I. Chapters 32–34, Theory of estimation. Chapters 35–37, Tests of significance, II.

MSC:
62-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics
62F10 Point estimation
62F03 Parametric hypothesis testing
62-02 Research exposition (monographs, survey articles) pertaining to statistics
01A75 Collected or selected works; reprintings or translations of classics
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