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

TL;DR

This paper establishes the optimal rates for estimating the stationary density of a stochastic differential equation from continuous data, revealing how anisotropic smoothness affects convergence and proposing adaptive kernel estimators.

Contribution

It characterizes the minimax rates for density estimation over anisotropic Hölder classes in high dimensions and introduces adaptive kernel methods that attain these rates.

Findings

Minimax rates depend on the ordering of smoothness parameters.

Kernel estimators achieve the optimal minimax rates.

Adaptive procedures are effective for both pointwise and integrated risks.

Abstract

We study the problem of the nonparametric estimation for the density $\pi$ of the stationary distribution of a $d$-dimensional stochastic differential equation $(X_t)_{t \in [0, T]}$. From the continuous observation of the sampling path on $[0, T]$, we study the estimation of $\pi(x)$ as $T$ goes to infinity. For $d\ge2$, we characterize the minimax rate for the $\mathbf{L}^2$-risk in pointwise estimation over a class of anisotropic H\"older functions $\pi$ with regularity $\beta = (\beta_1, ... , \beta_d)$. For $d \ge 3$, our finding is that, having ordered the smoothness such that $\beta_1 \le ... \le \beta_d$, the minimax rate depends on whether $\beta_2 < \beta_3$ or $\beta_2 = \beta_3$. In the first case, this rate is $(\frac{\log T}{T})^\gamma$, and in the second case, it is $(\frac{1}{T})^\gamma$, where $\gamma$ is an explicit exponent dependent on the dimension and $\bar{\beta}_3$, the harmonic mean of smoothness over the $d$ directions after excluding $\beta_1$ and $\beta_2$, the smallest ones. We also demonstrate that kernel-based estimators achieve the optimal minimax rate. Furthermore, we propose an adaptive procedure for both $L^2$ integrated and pointwise risk. In the two-dimensional case, we show that kernel density estimators achieve the rate $\frac{\log T}{T}$, which is optimal in the minimax sense. Finally we illustrate the validity of our theoretical findings by proposing numerical results.