Improved confidence set estimators of a multivariate normal mean and generalizations (Q1076453)

Let X be p-variate normal random variable with expectation \(\theta\) and known covariance matrix, assumed to be I(p\(\times p)\). The maximum likelihood estimator of \(\theta\) is \(\delta^{(0)}(X)=X\). The information contained in X can be used in several ways to construct a confidence set estimator for \(\theta\). Thus, the usual estimator is: \[ C_{\delta^{(0)}}=\{\theta:\quad | \theta -X| \leq c\} \] which is a p-ball centered at \(\delta^{(0)}(X)\), the radius of which is determined by the coefficient (1-\(\alpha)\). It is possible to improve on this estimator defining the improvement (dominance) as: \(C^*\) dominates C if (i) \(P\{\theta \in C^*\}\geq P\{\theta \in C\}\) and (ii) \(Vol(C^*)\leq Vol(C)\) with the strict inequality for a set of \(\theta\) or X. With this definition, the authors seek to improve on \(C_{\delta^{(0)}}\) by considering \(C_{\delta^{(a,\phi)}}\) with center at point estimator of the type \[ \delta^{(a,\phi)}=[1- \frac{a\phi (| X|)}{| X|^ 2}]_+X \] where \(y_+\) indicates the max(0,y). It should be noted that \(\delta^{(a,\phi)}\) is the positive part of the Baranchik estimator. Earlier work has shown that the confidence set estimator using \(\phi (| X|)=1\) dominates \(C_{\delta^{(0)}}\). The authors have examined here the sufficient conditions on \(\phi\) and a to achieve the dominance of \(C_{\delta^{(a,\phi)}}\) over \(C_{\delta^{(0)}}\). An extension of the normal case to spherically symmetric distribution case follows provided the relative increasing rate (RIR) is at least equal to that of the normal distribution. Numerous examples are provided with the tables of the coefficient a and coverage probabilities for the p-variate normal and the p-variate t.