Minimax rate of estimation for invariant densities associated to continuous stochastic differential equations over anisotropic Hölder classes

Chiara Amorino and Arnaud Gloter

Scandinavian Journal of Statistics, 2025, vol. 52, issue 1, 185-248

Abstract: We study the problem of the nonparametric estimation for the density π$$ \pi $$ of the stationary distribution of a d$$ d $$‐dimensional stochastic differential equation (Xt)t∈[0,T]$$ {\left({X}_t\right)}_{t\in \left[0,T\right]} $$. From the continuous observation of the sampling path on [0,T]$$ \left[0,T\right] $$, we study the estimation of π(x)$$ \pi (x) $$ as T$$ T $$ goes to infinity. For d≥2$$ d\ge 2 $$, we characterize the minimax rate for the L2$$ {\mathbf{L}}^2 $$‐risk in pointwise estimation over a class of anisotropic Hölder functions π$$ \pi $$ with regularity β=(β1,…,βd)$$ \beta =\left({\beta}_1,\dots, {\beta}_d\right) $$. For d≥3$$ d\ge 3 $$, our finding is that, having ordered the smoothness such that β1≤⋯≤βd$$ {\beta}_1\le \cdots \le {\beta}_d $$, the minimax rate depends on whether β2

Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:bla:scjsta:v:52:y:2025:i:1:p:185-248

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