The aim of the book is to show and explain applications of matrix derivatives in studying the limit behavior of some statistics used in multivariate statistical problems.
The book is divided into five chapters. Basic notions and assertions of classical matrix theory are recalled in Chapter 1. Chapter 2 focuses on block-matrices, Kronecker products and related material. Matrix derivatives in the sense of Neudecker [see J. R. Magnus and H. Neudecker, Matrix differential calculus with applications in statistics and econometrics. (1988; Zbl 0651.15001)] and E. C. MacRae [see Ann. of Statist. 2, 337-346 (1974; Zbl 0285.26013)] are introduced and their properties studied in Chapter 3. In Chapter 4, the MacRae type matrix derivatives are used to express in the cumulants, moments, and characteristic functions of random vectors and also the formal Edgeworth expansions of the corresponding distribution functions. Using the Neudecker type matrix derivatives the limit distributions of the sample mean, sample variance matrix and related statistics (Hotelling \(T^ 2\), eigenvalues, eigenvectors, etc.) are derived in Chapter 5.
The book is well-written and will be useful for advanced students, researchers in statistics as well as applied statisticians working in the area of multivariate statistical analysis.