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

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 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
$\nu.$

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