# James-Stein estimator with multiple samples

Let $$X_1, \dots, X_n \in \mathbb{R}^p$$ be i.i.d. samples from the $$p$$-normal distribution $$N(\theta, \tau^2 I)$$. Suppose we are interested in estimating $$\theta$$ with known variance $$\tau^2$$. Take the loss to be squared error.

Define the estimator $$\delta(X_1, \dots, X_n)$$. Is there someway to prove that a suitably constructed estimator $$\tilde{\delta}(\bar{X})$$ will never be dominated by $$\delta(X_1, \dots, X_n)$$, for the sample mean $$\bar{X} = \sum_{i=1}^n X_i / n$$? Some facts that might be helpful:

• The sample mean $$\bar{X}$$ is sufficient for $$\theta$$
• The loss is convex

I ask because often texts will only focus on the $$n=1$$ case when talking about the e.g. James-Stein estimator, suggesting that some averaging has occurred beforehand.

By the Rao-Blackwell theorem, any estimator $$\delta(X_1,\ldots,X_n)$$ can be improved (under any strictly convex loss) by its conditional expectation $$\tilde\delta(\bar X_n) = \mathbb E[\delta(X_1,\ldots,X_n)|\bar X_n]$$ since it keeps the same expectation but enjoys a smaller risk. To quote from the historical paper by Girshick and Savage (Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1951):
In particular it is known by a theorem of Blackwell (1951) that if the loss is quadratic in any function of $$\tau$$, only functions of [a sufficient statistic] $$u$$ can be admissible, and if any estimate is minimax there is a minimax function of $$u$$.