Estimation of a mean vector under quartic loss

Let X=(X1,…,Xp)X=(X1,…,Xp) be a p  -variate normal random vector with unknown mean θ=(θ1,…,θp)θ=(θ1,…,θp) and identity covariance matrix. Estimators δ=(δ1,…,δp)δ=(δ1,…,δp) of θθ are considered under the quartic loss ∑i=1p(δi-θi)4. For p⩾3p⩾3, we develop sufficient conditions on δ(X)=X+g(X)δ(X)=X+g(X) to improve upon the usual estimator δ0(X)=Xδ0(X)=X. To this end, we yield an unbiased estimator Og(X)Og(X) of the risk difference between δ(X)δ(X) and δ0(X)δ0(X). An interesting feature is that, to obtain adequate dominating estimators, Og(X)Og(X) is used in two ways. First, we search estimators such that Og(x)⩽0Og(x)⩽0 for any x∈Rpx∈Rp, which guarantees the desired domination. Then, to enlarge the class of improved estimators, we investigate conditions for which this inequality is satisfied in mean, that is, Eθ[Og(X)]⩽0Eθ[Og(X)]⩽0. In particular, no James–Stein estimator satisfies Og(X)<0Og(X)<0 for all X  , but Eθ[Og(X)]<0Eθ[Og(X)]<0 for p⩾5p⩾5 and a shrinkage factor 0