Improved estimation of the mean vector for Student-t model.

Abstract

Improved James-Stein type estimation of the mean vector $\mbox{\boldmath $\mu$}$ of a multivariate Student-t population of dimension p with $\nu$ degrees of freedom is considered. In addition to the sample data, uncertain prior information on the value of the mean vector, in the form of a null hypothesis, is used for the estimation. The usual maximum likelihood estimator (mle) of $\mbox{\boldmath $\mu$}$ is obtained and a test statistic for testing $H_0: \mbox{\boldmath $\mu$} = \mbox{\boldmath $\mu$}_0$ is derived. Based on the mle of $\mbox{\boldmath $\mu$}$ and the test statistic the preliminary test estimator (PTE), Stein-type shrinkage estimator (SE) and positive-rule shrinkage estimator (PRSE) are defined. The bias and the quadratic risk of the estimators are evaluated. The relative performances of the estimators are investigated by analyzing the risks under different conditions. It is observed that the PRSE dominates over the other three estimators, regardless of the validity of the null hypothesis and the value $\nu.$