Donald Richards (Pennsylvania State University)
Title: Goodness-of-Fit Testing for the Wishart Distributions
Abstract: The problem of testing that a random sample is drawn from a specific probability distribution is an old one, the most famous example perhaps being the problem of testing that a sequence of playing cards was drawn from a fairly shuffled deck. In recent years, random data consisting of positive definite (symmetric) matrices have appeared in areas of applied research such as factor analysis, diffusion tensor imaging, wireless communication systems, synthetic aperture radar, and models of financial volatility. Given a random sample of positive definite matrices, we develop a goodness-of-fit test for the Wishart distributions. We derive the asymptotic distribution of the test statistic in terms of a certain Gaussian random field, and we obtain an explicit formula for the corresponding covariance operator. The eigenfunctions of the covariance operator are determined explicitly, and the eigenvalues are shown to satisfy certain interlacing properties. As an application, we carry out a test that a financial data set has a Wishart distribution and, finally, we describe some recent research and open problems on related goodness-of-fit tests.