ROBUST ESTIMATION FOR ELLIPTICALLY SYMMET RICAL DISTRIBUTIONS WITH UNKNOWN COVARIANCE-MATRIX

Let X, V-1, ..., Vn-1 be n random vectors in R-p with joint density of the form f((Chi-theta)'Sigma(-1)(Chi-theta) + Sigma(j=1)(n-1) V-j' Si gma(-1) V-j), where both theta and Sigma are unknown. We consider the problem of estimating theta under the invariant loss (delta-theta)' Si gma(-1) (delta-theta) and propose estimators which dominate the usual estimator delta(0) (Chi) = Chi simultaneously for the entire class of such distributions. The proof involves the development of expressions which are analogous to unbiased estimators of risk and which in fact r educe to unbiased estimators of risk in the Gaussian case. The method is applicable to the case where Sigma is structured. (C) Academie des Sciences/Elsevier, Paris.