Source — A negative result on shrinkage estimators in small-N replication

rem:thm-31 rem:thm-31

Under the setup above, the $L^2$ risk of ${\hat{\theta }}_{2S}$ satisfies $$R_{2S}(\theta )\le d{\sigma }^2-\frac{(d-2{)}^2{\sigma }^4}{M+d{\sigma }^2},$$ with the inequality tight when $\hat{\mu }$ is a constant equal to $\theta$ (where both sides reduce to $2{\sigma }^2$).

lines 162–170 in main.tex