Shrinkage Estimation of a Location Parameter for a Multivariate Skew Elliptic Distribution

The multivariate skew elliptic distributions include the multivariate skew-t distribution, which is represented as a mean- and scale-mixture distribution and is useful for analyzing skewed data with heavy tails. In the estimation of location parameters in the multivariate skew elliptic distributions, we derive minimax shrinkage estimators improving on the minimum risk location equivariant estimator relative to the quadratic loss function. Especially in the skew-t distribution, we suggest specific improved estimators where the conditions for their minimaxity do not depend on the degrees of freedom. We also study the case of a general elliptically symmetrical distribution when the covariance matrix is known up to an unknown multiple, but a residual vector is available to estimate the scale.