Two-stage sequential estimation of a multivariate normal mean under quadratic loss (Q1074996)

Suppose \(Y_1,Y_2,\ldots,Y_ n\) are \(p\)-dimensional i.i.d. \(N(\theta,I)\) where the unknown \(\theta\) is to be estimated under quadratic loss \(L(\theta,\delta) = \|\delta-\theta\|^2\). The authors search two-stage sequential estimators which are better both in risk and sample size than the usual estimator of a given sample size \(n\). Let \(\bar X_ i = \sum^{i}_{j=1} Y_ j/i\). Stopping rules based on \(\|\bar X_ m\|\), where \(1\leq m\leq n-1\), and estimators of the form \[ \delta^ m_ c (Y_1, \ldots,Y_ n) = \begin{cases} \bar X_ m, \quad&\text{if \(\|\bar X_ m\| <c\),}\\ (1/an^{-1}\|\bar X_ n\|^{-2})\bar X_ n, \quad&\text{otherwise,} \end{cases} \] where \(0<a<2(p-2)\) are considered. It is shown that for \(p\geq3\), \(n\geq2\) there exists \(c^ m(a)>0\) such that for all \(0<c\leq c^ m(a)\) the estimators \(\delta^ m_ c\) have lower risks then \(\bar X_ n.\) Modifications of \(\delta^ m_ c\) replacing \(\bar X_ m\) by the corresponding James-Stein estimators and the case of unknown covariance matrix are also considered.