An introduction to order statistics
This book is an original instructional exposition of the general theory of order statistics, as an alternative to the well-known monographs due to \textit{H.A. David} [Order statistics. NY: Wiley (1970; Zbl 0223.62057)], \textit{J. Galambos} [The asymptotic threory of extreme order statistics. NY: Wiley (1978; Zbl 0381.62039)], \textit{B.C. Arnold} et al. [A first course in order statistics. NY: Wiley (1992; Zbl 0850.62008)], \textit{U. Kamps} [A concept of generalized order statistics. Stuttgart: Teubner (1995; Zbl 0851.62035)], \textit{V.B. Nevzorov} [Records. Mathematical theory. (Russian) Moscow: (2000; Zbl 0989.62029)]), among other treatments of this subject. The work brings together, in an elegant way, the most important results of this topic. The authors mainly refer to the case of independent and identically distributed random variables. The richness of theoretical results is supplemented by appropriate comments. The theory is grouped in 21 chapters, in an interesting presentation. Thus, each chapter begins with a short exposition of the basic definitions and fundamental results corresponding to the studied problem; then, the readers are urged to solve a lot of exercises. In fact, all significant theoretical developments and suggestive examples are contained in these sets of proposed exercises. The readers can find detailed solutions at the end of each chapter. These solutions are carefully written and filled with suitable remarks. Another feature of this paper is the attention given to the connection between the theory of order statistics and the theories of extreme values and record values, respectively. Also to mention the comprehensive treatment of the properties of estimators and predictors based on order statistics. The book is organized as follows. The first three chapters present the basic definitions of order statistics and related notions. The distributions of order statistics, sample quantiles and ranges of different kind are discussed. Chapter 4 proves various representations of order statistics in terms of sums or products of independent random variables. Here, the exponential and uniform distributions play a fundamental role. The conditional distributions of order statistics are studied in Chapter 5. The particularities of order statistics for discrete distributions are exposed in Chapter 6. The moments of order statistics are calculated in the next three chapters (with special attention given to the normal and the exponential distributions, respectively). Chapters 10 and 11 are devoted to the asymptotic behavior of the middle and intermediate order statistics and of the extreme order statistics (extreme value theory). Further, the authors present relevant results regarding the estimation of order statistics (Chapters 12--15). The last six chapters deal with various extensions of the concept, including the study of record values. The reading of the book is pleasant. The authors prove a considerable erudition and experience in this field.
scientific article; zbMATH DE number 6151705