Elements of large-sample theory.

*(English)*Zbl 0914.62001
Springer Texts in Statistics. New York, NY: Springer. xii, 631 p. (1999).

The purpose of the monograph under review is to make large-sample theory accessible to an audience with only two years of calculus and some linear algebra. To this end, difficult results are stated without proofs, although with clear statements of the conditions of their validity. In addition, the mode of probabilistic convergence used throughout is convergence in probability.

The book consists of seven chapters and an appendix. Chapter 1 (“Mathematical background”) provides a rigorous treatment of limits and other concepts that underlie all large sample theory. Chapter 2 (“Convergence in probability and in law”) covers the basic probabilistic tools: convergence in probability and in law, the central limit theorem, and the delta method. The next two chapters (“Performance of statistical tests” and “Estimation”) illustrate the application of these tools to hypothesis testing, confidence intervals, and point estimation, including efficiency comparisons and robustness considerations. In Chapter 5 (“Multivariate extensions”) the material of the first four chapters is extended to the multivariate case. Chapter 6 (“Nonparametric estimation”) is concerned with the extension of the earlier ideas to statistical functionals and, among other applications, provides introductions to U-statistics, density estimation, and the bootstrap. Chapter 7 (“Efficient estimators and tests”) deals with the construction of asymptotically efficient procedures, in particular, maximum likelihood estimators, likelihood ratio tests, and some of their variants. Finally, an Appendix briefly sketches several of more advanced topics which a reader of large-sample literature is likely to encounter. The topics are not included in the main text due to their more mathematical nature.

The book consists of seven chapters and an appendix. Chapter 1 (“Mathematical background”) provides a rigorous treatment of limits and other concepts that underlie all large sample theory. Chapter 2 (“Convergence in probability and in law”) covers the basic probabilistic tools: convergence in probability and in law, the central limit theorem, and the delta method. The next two chapters (“Performance of statistical tests” and “Estimation”) illustrate the application of these tools to hypothesis testing, confidence intervals, and point estimation, including efficiency comparisons and robustness considerations. In Chapter 5 (“Multivariate extensions”) the material of the first four chapters is extended to the multivariate case. Chapter 6 (“Nonparametric estimation”) is concerned with the extension of the earlier ideas to statistical functionals and, among other applications, provides introductions to U-statistics, density estimation, and the bootstrap. Chapter 7 (“Efficient estimators and tests”) deals with the construction of asymptotically efficient procedures, in particular, maximum likelihood estimators, likelihood ratio tests, and some of their variants. Finally, an Appendix briefly sketches several of more advanced topics which a reader of large-sample literature is likely to encounter. The topics are not included in the main text due to their more mathematical nature.

Reviewer: Joseph Melamed (Los Angeles)