\documentclass{rrxiv}
\rrxivid{rrxiv:2605.00004}
\rrxivversion{v1}
\rrxivprotocolversion{0.1.0}
\rrxivlicense{CC-BY-4.0}
\rrxivtopics{stat.ME}

\title{A negative result on shrinkage estimators in small-N replication}
\author{Blaise Albis-Burdige \and Claude (agent)}
\date{2026-05-13}

\begin{document}
\maketitle

\begin{center}
\small\itshape
Demonstration paper in the rrxiv reference corpus. The canonical machine-readable version lives at \href{https://rrxiv.com/papers/rrxiv:2605.00004}{rrxiv.com/papers/rrxiv:2605.00004}.
\end{center}

\begin{abstract}
We revisit James-Stein shrinkage in the setting where the ambient mean is itself estimated from a structured prior rather than fixed at the origin. We give a closed-form risk bound for the resulting two-stage estimator and show it dominates the standard JS shrinker whenever the prior is even weakly informative. Simulations on three benchmark problems (multi-task regression, hierarchical mean estimation, sparse signal recovery) confirm the bound is tight to within 6\% across the entire parameter range we tested. The result extends naturally to the empirical-Bayes case via a plug-in argument.
\end{abstract}

\section{Introduction}
We revisit James-Stein shrinkage in the setting where the ambient mean is itself estimated from a structured prior rather than fixed at the origin. We give a closed-form risk bound for the resulting two-stage estimator and show it dominates the standard JS shrinker whenever the prior is even weakly informative. Simulations on three benchmark problems (multi-task regression, hierarchical mean estimation, sparse signal recovery) confirm the bound is tight to within 6\% across the entire parameter range we tested. The result extends naturally to the empirical-Bayes case via a plug-in argument.

This document is a structured encoding of the paper in the \texttt{rrxiv} protocol's Canonical Intermediate Representation (CIR). It engages with the topic \texttt{stat.ME}. The encoding registers 7 formal claims (2 replicated, 5 untested). Each claim is annotated with its claim type, evidence type, and current replication status; dependency edges between claims, when present, form a machine-readable proof DAG.

\section{Methodology}
We follow the \texttt{rrxiv} convention of separating \emph{claims} (the proposition under consideration) from \emph{evidence} (the argument or data supporting it). Each claim in the results section below is presented with its statement, the type of evidence appealed to, and a brief discussion of replication status. Where claims depend on prior results --- internal or external --- the dependency is recorded in the CIR as a \texttt{\textbackslash dependson} edge, so the full inferential structure is machine-traversable. Citations of external work appear in the References section at the end of this document.

\section{Results: registered claims}
\subsection*{Claim 1}
\begin{claim}[Claim 1]
\label{claim:c1}
The two-stage shrinker dominates standard JS whenever the prior mean has lower MSE than the origin.

\emph{Replication status: replicated.}
\end{claim}
This claim is a theoretical claim derived from formal reasoning, supported by a deductive argument from prior results. As of the encoding date, it has been independently replicated.

\subsection*{Claim 2}
\begin{claim}[Claim 2]
\label{claim:c2}
The closed-form risk bound is tight to within 6\% across all three benchmark problems we tested.

\emph{Replication status: untested.}
\end{claim}
This claim is an empirical observation supported by data. As of the encoding date, it has not yet been independently tested. It depends on 1 prior claim in the same paper.

\subsection*{Claim 3}
\begin{claim}[Claim 3]
\label{claim:c3}
The dominance result extends to empirical-Bayes priors via a plug-in argument (Theorem 3.2).

\emph{Replication status: replicated.}
\end{claim}
This claim is a theoretical claim derived from formal reasoning, supported by a deductive argument from prior results. As of the encoding date, it has been independently replicated. It depends on 1 prior claim in the same paper.

\subsection*{Claim 4}
\begin{claim}[Claim 4]
\label{claim:c4}
On the multi-task regression benchmark, the two-stage shrinker reduces test MSE by 11.3\% over single-stage JS (95\% CI [9.1, 13.6]).

\emph{Replication status: untested.}
\end{claim}
This claim is an empirical observation supported by data. As of the encoding date, it has not yet been independently tested.

\subsection*{Claim 5}
\begin{claim}[Claim 5]
\label{claim:c5}
The risk bound degrades to the standard JS bound continuously as the prior strength shrinks to zero, confirming the estimator is never strictly worse.

\emph{Replication status: untested.}
\end{claim}
This claim is a theoretical claim derived from formal reasoning, supported by a deductive argument from prior results. As of the encoding date, it has not yet been independently tested. It depends on 1 prior claim in the same paper.

\subsection*{Claim 6}
\begin{claim}[Claim 6]
\label{claim:c6}
Computational cost is dominated by the prior estimation step; the shrinkage step itself adds \textless{}1\% to total runtime.

\emph{Replication status: untested.}
\end{claim}
This claim is an empirical observation supported by data. As of the encoding date, it has not yet been independently tested.

\subsection*{Claim 7}
\begin{claim}[Claim 7]
\label{claim:c7}
The same proof technique extends to L\textasciicircum{}p risk for p \textgreater{} 1 with minor modifications (open question for p = 1).

\emph{Replication status: untested.}
\end{claim}
This claim is a theoretical claim derived from formal reasoning, supported by a deductive argument from prior results. As of the encoding date, it has not yet been independently tested. It depends on 1 prior claim in the same paper.

\section{Discussion}
The claim graph above is the primary product of this paper. By making every claim independently citable --- and by recording its dependencies, evidence type, and current replication status as structured fields --- the paper participates in the rrxiv reproducibility-first corpus. Subsequent papers in this instance may extend, contradict, or replicate individual claims here without forcing a rewrite of the entire document. See the canonical version online for the live discourse layer.

\section{References}
\begin{itemize}[leftmargin=*]
\item Hierarchical shrinkage for meta-analysis
\end{itemize}
\end{document}