Local likelihood estimation of local parameters for nonstationary random fields

TL;DR

This paper introduces a weighted local likelihood method for estimating local parameters in nonstationary random fields, effectively handling irregular sampling and balancing bias-variance trade-offs.

Contribution

It presents a novel local likelihood estimation technique that adapts to nonstationarity and irregular sampling, with a method for bandwidth selection and demonstration through simulations.

Findings

Improved estimation accuracy over naive methods

Effective handling of irregular sampling locations

Successful simulation results for local smoothness estimation

Abstract

We develop a weighted local likelihood estimate for the parameters that govern the local spatial dependency of a locally stationary random field. The advantage of this local likelihood estimate is that it smoothly downweights the influence of far away observations, works for irregular sampling locations, and when designed appropriately, can trade bias and variance for reducing estimation error. This paper starts with an exposition of our technique on the problem of estimating an unknown positive function when multiplied by a stationary random field. This example gives concrete evidence of the benefits of our local likelihood as compared to na\"ive local likelihoods where the stationary model is assumed throughout a neighborhood. We then discuss the difficult problem of estimating a bandwidth parameter that controls the amount of influence from distant observations. Finally we present a simulation experiment for estimating the local smoothness of a local Mat\'ern random field when observing the field at random sampling locations in $[0,1]^2$.