ESTIMATION OF NORMAL COVARIANCE AND PRECISION MATRICES WITH INCOMPLETE DATA

Suppose that we have n independent observations from N(p)(0, SIGMA) and, in addition, we have m independent observations available on the first q(q < p) coordinates. Assuming that X(i)'s and Y(i)'s are independent, we consider the problem of estimation of SIGMA and SIGMA-1 respectively under the loss functions L(SIGMA, SIGMA-triple-overdot) = tr(SIGMA-triple-overdot-SIGMA-1) - log\SIGMA-triple-overdot-SIGMA-1\ - p and L1(SIGMA-1, SIGMA-triple-overdot-1) = tr(SIGMA-triple-overdot-1-SIGMA) - log\SIGMA-triple-overdot-1-SIGMA\ - p. We propose some new estimators that dominate the best lower triangular invariant minimax estimator under the loss L. We also derive the best lower triangular invariant minimax estimator of SIGMA-1 under L1 and suggest some estimators that dominate it.