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Abstract:  This talk is based on joint work with Elena Hadjicosta.In recent years, random data consisting of positive definite (symmetric) matrices have appeared in several areas of applied research, e.g., diffusion tensor imaging, wireless communication systems, synthetic aperture radar, and volatility models in finance. Given a random sample of such matrices, we wish to test whether the data are drawn from a given statistical population. In this talk, we apply the Hankel transform of matrix argument to develop goodness-of-fit tests for the Wishart distributions. The asymptotic distribution of the test statistic is derived in terms of the integrated square of a Gaussian random field, and an explicit formula is obtained for the corresponding covariance operator. The eigenfunctions of the covariance operator are determined explicitly, and the eigenvalues are shown to satisfy certain interlacing properties. 

Throughout this work, the Bessel functions of matrix argument of Herz (1955) and the zonal polynomials of James (1964) play a crucial role. Also, our results raise the possibility of extending to the Wishart distributions many statistical properties of the classical, one-dimensional gamma distributions.