Statistical inference for a stochastic partial differential equation related to an ecological niche

TL;DR

This paper develops statistical inference methods for a stochastic PDE model of ecological populations, deriving estimators for key parameters and validating them through simulations and theoretical proofs.

Contribution

It introduces a novel approach to estimate parameters of a population density SPDE using Galerkin projection and proves estimator properties.

Findings

Maximum likelihood estimators are consistent.

Estimators are asymptotically normal.

Method validated with numerical simulations.

Abstract

In this paper, we use a stochastic partial differential equation (SPDE) as a model for the density of a population under the influence of random external forces/stimuli given by the environment. We study statistical properties for two crucial parameters of the SPDE that describe the dynamic of the system. To do that we use the Galerkin projection to transform the problem, passing from the SPDE to a system of independent SDEs; in this manner, we are able to find the Maximum likelihood estimator of the parameters. We validate the method by using simulations of the SDEs. We prove consistency and asymptotic normality of the estimators; the latter is showed using the Malliavin-Stein method. We illustrate our results with numerical experiments.