Wishart’s Paper

Abstract

Some of the earlier writers, notably J. Wishart and R. A. Fisher, made extensive use of geometric reasoning in order to compute the volume elements that arise from changes of variable. We have selected the paper by Wishart (1928) to illustrate this type of argument. The Wishart density function so obtained plays a central role in the discussions which follow in later chapters, where different methods of obtaining density functions of maximal invariants are discussed. See especially Chapters 5, 10 and 11. Section 4.1 gives Wishart’s argument. Section 4.2 explains an idea of James (1955a) by means of which the noncentral Wishart density may be obtained. The expression for the density function involves an integral over O(n) which cannot be evaluated in closed form by any known method. Section 4.3 illustrates the difficulty that one encounters when trying to obtain a power series expansion of the integral. This difficulty led to the development of zonal polynomials and the series for hypergeometric functions, mostly the work of A. T. James (see Chapters 12 and 13). The problems, Section 4.4, leave certain necessary details to the reader, with hints. Material relevant to some of the problems will be found in other chapters.