Stein-type improvements of confidence intervals for the generalized variance

Abstract

Based on independent random matices X: p×m and S: p×p distributed, respectively, as N _pm (μ, ∑ ⊗ I _m ) and W _p (n, ∑) with μ unknown and n≥p, the problem of obtaining confidence interval for |∑| is considered. Stein's idea of improving the best affine equivariant point estimator of |∑| has been adapted to the interval estimation problem. It is shown that an interval estimator of the form |S|(b ⁻¹, a ⁻¹) can be improved by min{|S|, c|S +XX'|}(b ⁻¹, a ⁻¹) for a certain constant c depending on (a, b).