Parameter estimation for fractional stochastic heat equations : Berry-Esséen bounds in CLTs

Soukaina Douissi, Fatimah Alshahrani
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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.
分数随机热方程的参数估计 :CLT中的贝里-埃森边界
这项工作的目的是利用最大似然估计法(MLE)估计由时空噪声加法驱动的分式热方程的漂移系数。在论文的第一部分,在有限的时间区间 $[0, T ]$ 内连续观测解的前 $N$Fourier 模式。当 $N \rightarrow \infty$ 和/或 $T \rightarrow \infty$ 时,为 MLE 的中心极限定理提供了明确的 Wasserstein 距离上限。而在本文的第二部分,$N$傅立叶模式是在统一时间网格下观察到的:$t_i = i \frac{T}{M}$,$i=0,...,M,$ 其中$M$是时间网格点的数量。研究了当 $T,M,N \rightarrow + \infty$ 时的一致性和渐近正态性,以及 CLT 中的收敛速率规律。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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