Summary: Let \(X=(X_1,\dots,X_p)\) be a \(p\)-variate normal random vector with unknown mean \(\theta=(\theta_1,\dots,\theta_p)\) and identity covariance matrix. Estimators \(\delta=(\delta_1,\dots,\delta_p)\) of \(\theta\) are considered under the quartic loss \(\sum_{i=1}^p (\delta_i-\theta_i)^4\). For \(p\geq 3\), we develop sufficient conditions on \(\delta(X)=X+g(X)\) to improve upon the usual estimator \(\delta^0(X)=X\). To this end, we yield an unbiased estimator \({\mathcal O}g(X)\) of the risk difference between \(\delta(X)\) and \(\delta^0(X)\).
An interesting feature is that, to obtain adequate dominating estimators, \({\mathcal O}g(X)\) is used in two ways. First, we search estimators such that \({\mathcal O}g(x)\leq 0\) for any \(x\in\mathbb R^p\), which guarantees the desired domination. Then, to enlarge the class of improved estimators, we investigate conditions for which this inequality is satisfied in the mean, that is, \(E_\theta[{\mathcal O}g(X)]\leq 0\). In particular, no James-Stein estimator satisfies \({\mathcal O}g(X)<0\) for all \(X\), but \(E_\theta[{\mathcal O}g(X)]<0\) for \(p\geq 5\) and a shrinkage factor \(0<a\leq2(p-4)\).