Affine shrinkage estimator for a binomial proportion
= Affine shrinkage estimator for a binomial proportion
For $X\sim\operatorname{Bin}(n,\theta)$ and
$$
\widetilde\theta=wX/n+(1-w)\theta_0,
$$
the mean squared error is
$$
\frac{w^2}{n}\theta(1-\theta)
+(1-w)^2(\theta-\theta_0)^2.
$$
When $\theta_0=1/2$, its maximal risk is at $\theta=1/2$ or at an endpoint of $[0,1]$.