Improved Estimation of Generalized Variance and Precision

Abstract

Consider a multivariate normal random matrix X, where X (p × r) is distributed with independent columns X _{ i }\sim N _{ p }\left ( \zeta _{ i },\sum \right ) , i = 1, 2, …, r, and S has Wishart distribution W p(n, Σ). Estimators of |Σ| and |Σ−1| based on X and S are derived under the loss function L \left (\left |\sum \right |,\left |\circ {>\sum }\right. \right )=\left |\circ {>\sum }\sum ^{-1}\right |+\left |\sum \circ {>\sum }^{-1}\right |-2 . The risks of the improved estimators are evaluated in terms of an incomplete beta function of matrix argument. Numerical results for the risk of the improved estimator for selected values of r, p and n indicate a reduction in risk over the best affine equivariant estimator c 0|S|, where for n > p +1, c _{ o }=\left {\left (\left ( n - p -1\right )!\left ( n - p \right )!\right )/\left ( n -2\right )! n !\right }^{1/2} .