Robustness of statistical models

TL;DR

This paper investigates the stability of certain statistical structures on smooth manifolds, demonstrating their robustness under small perturbations using advanced mathematical theorems, particularly for structures induced by standard linear models.

Contribution

It proves the robustness of generic statistical structures on manifolds induced by standard linear models using the Nash--Gromov implicit function theorem.

Findings

Robustness of statistical structures on manifolds is established.

The results apply to structures induced by standard linear models on high-dimensional spaces.

The approach uses advanced differential geometric techniques.

Abstract

A statistical structure $(g, T)$ on a smooth manifold $M$ induced by $(\tilde M, \tilde g, \tilde T)$ is said to be {\em robust} if there exists an open neighborhood of $(g,T)$ in the fine $C^{\infty}$-topology consisting of statistical structures induced by $(\tilde M, \tilde g, \tilde T)$. Using Nash--Gromov implicit function theorem, we show robustness of the generic statistical structure induced on $M$ by the standard linear statistical structure on ${\R}^N$, for $N$ sufficiently large.