A negative result on shrinkage estimators in small-N replication

Abstract

We give a closed-form $L^2$ risk bound for a two-stage James-Stein (JS) shrinker whose target is itself an estimate from a structured prior, and prove the resulting estimator dominates the classical JS shrinker whenever the prior mean has lower mean squared error than the origin. The dominance extends to empirical-Bayes plug-in priors and degrades continuously to standard JS as the prior strength tends to zero. The result is mathematically positive but operationally negative for the small-$N$ replication context the method is most often recommended for: in three benchmarks and a multi-task regression study, the cost of estimating the prior dominates the gain unless the number of cross-replication groups exceeds roughly thirty. We argue this is the regime where the recommendation in the methodological literature should be reversed.