= Solution
For
$$
\widetilde\theta=w\frac Xn+(1-w)\theta_0,
$$
the bias and variance are
$$
\mathbb E_\theta\widetilde\theta-\theta
=(1-w)(\theta_0-\theta),
\qquad
\operatorname{Var}_\theta(\widetilde\theta)
=\frac{w^2}{n}\theta(1-\theta).
$$
The <bias-variance decomposition of mean squared error> yields
$$
\boxed{
g_{w,\theta_0}(\theta)
=\frac{w^2}{n}\theta(1-\theta)
+(1-w)^2(\theta-\theta_0)^2
}.
$$
This is the risk formula for an <affine shrinkage estimator for a binomial proportion>.
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