ON ESTIMATING EIGENVALUES OF THE SCALE MATRIX OF THE MULTIVARIATE-F DISTRIBUTION

Assume that a p X p positive definite random matrix F follows the multivarite F distribution with a scale matrix DELTA. The main concern of this paper is of estimating eigenvalues of DELTA and the merit of estimates is evaluated by a loss that is a function of DELTA and an orthogonally invariant estimator DELTA(F) since eigenvalues of DELTA(F) are taken as estimates of eigenvalues of DELTA. By recursive use of integration by parts formula on this distribution, we show that the risk of the orthogonally invariant estimator due to Gupta and Krishnamoorthy (1987) is smaller than minimax risk when p = 2 so that it is minimax.