James-Stein-style estimator when we place greater importance on some components The James-Stein estimator allows us to get a better overall estimate of a mean vector (length $\ge 3$) than we would be able to get by estimating the components independently. My intuition is that, under the hood, it is willing to sacrifice some of the components in order to tighten up estimates of the others, resulting in tighter estimates overall.
This sounds nice, but it makes the assumption that we would be willing to sacrifice any component for any other, that the cost of missing one bad is offset by tightening up another estimate. This need not be the case. Further, I do not see a way that such an estimator makes clear which components are sacrificed, so I would not even know which marginal estimates are the best.
What happens when we want to emphasize one component more than the others or want to know which components are sacrificed for which others?

If one wishes to favour the estimation of one component of the mean vector, the loss function can penalise the attached error more for that component. Using a quadratic loss,
$$\text{L}(\theta,\delta)=(\delta-\theta)^\text{T}Q(\delta-\theta)$$
where $Q$ is a symmetric positive semi-definite matrix, there exist James-Stein estimators that dominate the MLE $\varphi_0(X)$ for that loss, provided $Q$ is of rank $3$ or more, as e.g.
$$\varphi(x)=\varphi_0(x)-h(\varphi_0(x)^\text{T}B\varphi_0(x))C\varphi_0(x)$$
where $B$ is a symmetric symmetric positive definite matrix and $C$ is a symmetric matrix such that $B$ and $C$ share an eigenbasis. See our 1989 paper for details (under some conditions on the function $h$, obviously).
