STEIN-TYPE IMPROVEMENTS OF CONFIDENCE INTERVALS FOR THE GENERALIZED VARIANCE

SANAT K. SARKAR

Department of Statistics, Temple University, Philadelphia, PA 19122, U.S.A.

(Received October 24, 1989; revised March 19, 1990)

Abstract. Based on independent random matices X: p × m and S: p × p distributed, respectively, as N_{pm}(μ, Σ ⊗ I_{m}) and W_{p}(n, Σ) with μ unknown and n ≥ p, the problem of obtaining confidence interval for |Σ| is considered. Stein’s idea of improving the best affine equivariant point estimator of |Σ| 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).

Key words and phrases: Generalized variance, invariant interval estimators, Stein-type improvements.