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

Improved James-Stein type estimation of the mean vector$\mbox{\boldmath $\mu$}$ of a multivariate Student-t population ofdimension 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 ofa null hypothesis, is used for the estimation. The usual maximum likelihoodestimator (mle) of $\mbox{\boldmath $\mu$}$ is obtained and a test statisticfor testing $H_0: \mbox{\boldmath $\mu$} = \mbox{\boldmath $\mu$}_0$ isderived. Based on the mle of $\mbox{\boldmath $\mu$}$ and the test statisticthe preliminary test estimator (PTE), Stein-type shrinkageestimator (SE) and positive-rule shrinkage estimator (PRSE) aredefined. The bias and the quadratic risk of the estimators areevaluated. The relative performances of theestimators are investigated by analyzing the risks under differentconditions. It is observed that the PRSE dominates over the other threeestimators, regardless of the validity of the null hypothesis and the value$\nu.$