Simultaneous Reconstruction of Shuffled Histograms from a Retained Statistic: Exact leading coefficients for finite hypothesis families

Abstract

Le Cam deficiency is the largest increase in Bayes risk caused by retaining a statistic, over all priors and decision problems with losses in $[0,1]$. We compute its exact leading term for finite hypothesis families in growing shuffle models. For a bounded correlated message bundle in an iid categorical background with an observed finite tag, the coefficient is a convex program over exposed faces with Gaussian-polynomial scores; its order is set by the first differing conditional factorial-moment degree. Tag-preserving kernels can be suboptimal, and an exact example has family coefficient $4/3$ times the largest pairwise coefficient. For a critical rare-event channel along $m+1$ consecutive compositions, retaining a distinguished-symbol count has loss $n^{-1/2}\kappa \phi (0){\mathcal{V}}_m(\mu )+o(n^{-1/2})$, where $\mu =\theta \lambda$, ${\kappa }^2={\chi }^2(q\Vert p)/(1-\theta )>0$, and $\phi$ is the standard normal density. We evaluate ${\mathcal{V}}_2$ for all $\mu >0$. For every $m\ge 2$, the full-chain value $e^{-\mu }(1-1/{\sum }_{k=0}^m{\mu }^k)$ is exact if and only if ${\sum }_{k=2}^{m+1}{\mu }^k\le 2$; an explicit large-mean regime is attained by the endpoint pair. Exact-arithmetic computer-assisted certificates give the complete four-composition phase diagram. At $\mu =\sqrt{2}$, the ratio to the largest pairwise coefficient is strictly below two for every finite window and tends to two as $m\to \infty$. A universal upper bound of two and Hypothesis P, which asserts that admissible vertex paths attain every window coefficient, remain open outside the proved regimes. The comparison principle is classical; the contributions are the model-specific scores, exact coefficients, and certificates.

arXiv comment36 pages; ancillary files (anc/) contain exact-arithmetic verifiers. v2 is a new paper that replaces v1 entirely (new title, abstract and content). v1, "Universal Shuffle Asymptotics, Part III" (numerical companion on Zenodo, DOI:10.5281/zenodo.18978051), remains available as version 1