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On estimating eigenvalues of the scale matrix of the multivariate \(F\) distribution. (English) Zbl 0781.62072

Summary: Assume that a \(p\times p\) positive definite random matrix \(F\) follows the multivariate \(F\) distribution with a scale matrix \(\Delta\). The main concern of this paper is estimating eigenvalues of \(\Delta\) and the merits of estimates are evaluated by a loss that is a function of \(\Delta\) and an orthogonally invariant estimator \(\widehat{\Delta}(F)\) because the eigenvalues of \(\widehat{\Delta}(F)\) are taken as estimates of eigenvalues of \(\Delta\). By recursive use of integration by parts formulas on this distribution, we show that the risk of the orthogonally invariant estimator due to A. K. Gupta and K. Krishnamoorthy [Tech. Rep. No. 87-11, Bowling Green State Univ. (1987)] is smaller than the minimax risk when \(p=2\) so that it is minimax.

MSC:

62H12 Estimation in multivariate analysis
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