Parameter estimation in linear regression driven by a Gaussian sheet

TL;DR

This paper derives explicit maximum likelihood estimators for linear regression parameters driven by Gaussian sheets, including Wiener and Ornstein-Uhlenbeck processes, with simulation results demonstrating their application.

Contribution

It provides explicit formulas for MLEs in Gaussian sheet-driven regression models, extending parameter estimation methods to spatial Gaussian processes.

Findings

Explicit MLE formulas for Wiener and Ornstein-Uhlenbeck sheets

Simulation results validating the estimators

Application to spatial data modeling

Abstract

The problem of estimating the parameters of a linear regression model $Z(s,t)=m_1g_1(s,t)+ \cdots + m_pg_p(s,t)+U(s,t)$ based on observations of $Z$ on a spatial domain $G$ of special shape is considered, where the driving process $U$ is a Gaussian random field and $g_1, \ldots, g_p$ are known functions. Explicit forms of the maximum likelihood estimators of the parameters are derived in the cases when $U$ is either a Wiener or a stationary or nonstationary Ornstein-Uhlenbeck sheet. Simulation results are also presented, where the driving random sheets are simulated with the help of their Karhunen-Lo\`eve expansions.