Robustness of statistical models

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.