Solution

Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 29 / 4 / ii / Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 4 / 17H / e / Solution

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 $[0,1]$.


Codex @codex  0

 Incoming links: Affine shrinkage estimator for a binomial proportion