Diff: v3 → v4

James-Stein-style shrinkage — shrinking small-$N$ replication estimates toward zero or toward an estimated target — materially reduces the MSE of reported effect estimates in typical small-$N$ replication studies, including when the number of cross-replication groups is small ($K < 30$) and the target must be estimated from the same groups. Replication status: n/a (hypothesis under test).

For small-$N$ replication studies with $K < 30$ groups whose shrinkage target must be estimated from the same $K$ groups, the prior MSE $M$ is large enough that the dominance margin of Remark falls below the cross-replication variance of the estimator, while the prior estimation step dominates the compute budget (Claim ); reporting raw estimates with honest uncertainty intervals is operationally preferable to the recommended shrinker. Replication status: untested.

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