APPROXIMATING BY THE WISHART DISTRIBUTION

TÕNU KOLLO^{1} AND DIETRICH VON ROSEN^{2}

1Department of Mathematical Statistics, University of Tartu, J. Liivi Street 2, EE-2400 Tartu, Estonia

2Department of Mathematics, Uppsala University, Box 480, S-751 06 Uppsala, Sweden

(Received January 24, 1994; revised February 6, 1995)

Abstract. Approximations of density functions are considered in the multivariate case. The results are presented with the help of matrix derivatives, powers of Kronecker products and Taylor expansions of functions with matrix argument. In particular, an approximation by the Wishart distribution is discussed. It is shown that in many situations the distributions should be centred. The results are applied to the approximation of the distribution of the sample covariance matrix and to the distribution of the non-central Wishart distribution.

Key words and phrases: Density approximation, Edgeworth expansion, multivariate cumulants, Wishart distribution, non-central Wishart distribution, sample covariance matrix.