Parameter estimation for fractional stochastic heat equations : Berry-Ess\'een bounds in CLTs

TL;DR

This paper develops statistical methods for estimating the drift in fractional heat equations driven by noise, providing explicit bounds for the convergence rates of the maximum likelihood estimator under various observation schemes.

Contribution

It introduces explicit Wasserstein distance bounds for the CLT of the MLE and analyzes the estimator's asymptotic properties under different observation regimes.

Findings

Explicit Wasserstein bounds for CLT of MLE when N or T go to infinity.

Consistency and asymptotic normality of the MLE under increasing N, T, and M.

Rates of convergence in law for the estimator.

Abstract

The aim of this work is to estimate the drift coefficient of a fractional heat equation driven by an additive space-time noise using the Maximum likelihood estimator (MLE). In the first part of the paper, the first $N$ Fourier modes of the solution are observed continuously over a finite time interval $[0, T ]$. The explicit upper bounds for the Wasserstein distance for the central limit theorem of the MLE is provided when $N \rightarrow \infty$ and/or $T \rightarrow \infty$. While in the second part of the paper, the $N$ Fourier modes are observed at uniform time grid : $t_i = i \frac{T}{M}$, $i=0,..,M,$ where $M$ is the number of time grid points. The consistency and asymptotic normality are studied when $T,M,N \rightarrow + \infty$ in addition to the rate of convergence in law in the CLT.