Rao-Blackwell estimator of a squared normal mean
= Rao-Blackwell estimator of a squared normal mean
{c}
For independent $X_1,X_2\sim N(\mu,1)$, the unbiased estimator $X_1^2-1$ of $\mu^2$ has mean square error $2+4\mu^2$. Conditioning on the sufficient statistic $T=X_1+X_2$ gives
$$
\mathbb E[X_1^2-1\mid T]=\frac{T^2}{4}-\frac12,
$$
whose mean square error is $1/2+2\mu^2$.