Mathematical statistics. Basic ideas and selected topics. Volume I. 2nd ed. (English) Zbl 1380.62002

Chapman & Hall/CRC Texts in Statistical Science Series. Boca Raton, FL: CRC Press (ISBN 978-1-4987-2380-0/book+ebook). xxiv, 556 p. (2015).
An overview of developments in statistics since the last edition (1977; Zbl 0403.62001) is presented.
The development of devices using novel instrument and scientific techniques has shifted away from small samples to methods for inference based on large numbers of observations and minimal assumptions: asymptotic methods in semiparametric models, models with infinite numbers of parameters, time series, temporal spatial series, methods for inference for bootstrap and Markov chains, the techniques not describable in closed mathematical form, but rather through elaborate algorithms for which the existence of solutions are important, the interplay between the number of observations and the number of parameters of a model, the appropriate theories, the extensive development of graphical methods.
The goal of the book is the description of basic concepts of mathematical statistics relating theory to practice.
Volume I presents the basic classical concepts at the Ph.D. level without using measure theory.
In Chapter 1, the data structure is visualized by simple examples used in the next chapters. Decision theory is used to establish if the proposed models generating the data are adequate for the purpose of the experiment.
The topics include the methods of estimation (Chapter 2): prediction, testing and confidence sets, Bayesian analysis, the more general approach of decision theory and the treatment of maximum likelihood estimates (MLEs) in canonical \(k\)-parameters exponential families.
Appendices A–B include: basic probability, some classical discrete distribution, respectively more advanced topics such as multivariate Gaussian distribution, weak convergence in Euclidean spaces, probability inequalities as Chauchy-Schwartz, Chebychev, information, Hoeffding, Hölder, Jensen, Markov, Shannon.
Some classical discrete distributions: the binomial distribution \(B\left( n,p\right)\), corresponding to an experiment with only two possible outcomes: \(S\) (success) and \(F\) (failure) having \(P\left( S\right) =p\) as Bernoulli trial with probability \(p\).
Repeating such an experiment and independently we say to perform Bernoulli trials, with success probability \(p\).
Chapters 3 and 4 are on the theory of testing and confidence regions, including some optimality theory for estimation and elementary robustness considerations.
Chapter 5 is devoted to basic asymptotic approximations with one-dimensional parameter models as examples. It includes proofs of consistency and asymptotic normality and optimality of maximum likelihood procedures in inference. A section relating Bayesian and frequents inference via the Bernstein-von Mises theorem is presented.
Chapter 6 is devoted to inference in multivariate and multiparameter models and includes some parallels to optimality theory and comparisons of the Bayes and frequentist procedures. Like the spirit of the book, examples of generalized linear models are presented including the binary logistic regression.
The references are up to date and very comprehensive in all chapters. The Appendices A and B are very rich and useful for students and researchers.
For Volume II see [Zbl 1397.62003].


62-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics