Prove that risk function is analytic?

Question

I'm considering the statistical minimax estimation problem of the bounded normal mean:

Specifically, the problem is to find the minimax estimator of $X \sim N(\theta,1)$ where $\theta \in [-\tau,\tau]$. The loss function is squared loss $l(\delta(X),\theta) = (\delta(X) - \theta)^2$.

A lot of results I am finding online note that it is "well-known" that the risk function $R$ given by:

$$R(\theta) = E[(E[\theta|X]-\theta)^2]$$

is analytic for any prior distribution on $\theta$. How would I go about showing this?

Answer 1

although this may be a little late to respond I think your question is really interesting. Casella and Strawderman have shown that "if $\tau < 1/\sqrt{n}$ then $\tau.\tanh(\tau n \bar{X})$ is minimax". Ref. https://projecteuclid.org/download/pdf_1/euclid.aos/1176345527. Also you can go through pages 327-329 and 394-395 (problem no. 2.9 in chapter 5) in Theory of Point Estimation by Lehmann and Casella.