Improving the James-Stein estimator via q-number shrinkage

Abstract

For estimating the mean vector of a $p$-variate normal distribution ($p\ge 3$) under quadratic loss, the positive-part James-Stein estimator dominates the original Jame-Stein estimator though it suffers from a non-smooth thresholding boundary. In this paper, we first express the shrinkage factor of the positive-part estimator using an infinite series. Truncating this series at a finite degree $m$ allows us to interpret the factor as a special case of $q$-numbers. Leveraging these $q$-numbers parameterized by a positive real index $m$, we propose a novel smooth estimator. We prove that the proposed estimator dominates the original James-Stein estimator for any dimension $p\ge 3$, provided the truncation parameter satisfies $m\ge 0.95\sqrt{p-2}$.

arXiv commentMajor revision with q-number formulation; title updated