Random volumes under a general matrix-variate model

The convex hull generated by p linearly independent points in Euclidean n-space, n >= p will almost surely determine a p-simplex and the corresponding p-parallelotope. The volume of this p-parallelotope is v = vertical bar XX'vertical bar(1)/(2) where the rows of the p x n, n >= p matrix of rank p represent the p linearly independent points. If the points are random points in some sense then v becomes a random volume. The distribution of this random volume v when the matrix X has a very general real rectangular matrix-variate density is the topic of this paper. The complicated classical procedures based on integral geometry techniques for dealing with such problems are replaced by a simpler procedure based on Jacobians of matrix trails formations and functions of matrix argument. Apart from the distribution of v under this general model, arbitrary moments of v, connection to the likelihood ratio statistic or lambda-criterion for testing hypotheses on the parameters of multivariate normal distributions, connections to Mellin-Barnes integrals and Meijer's G-function, connection to the concept of generalized variance, various structural decompositions of v and special cases are also examined here. (c) 2007 Elsevier Inc. All rights reserved.