# On identifiability and consistency of the nugget in Gaussian spatial process models

## TL;DR

This paper investigates the theoretical properties of the 'nugget' parameter in Gaussian spatial models, proving its identifiability and the consistency of its estimators under in-fill asymptotics, with supporting simulations.

## Contribution

It extends fixed domain asymptotic results to include the nugget, establishing parameter identifiability and estimator consistency in Gaussian spatial models.

## Key findings

- Nugget parameter is identifiable under in-fill asymptotics.
- Maximum likelihood estimators for the Matérn covariance parameters are consistent.
- Simulation studies demonstrate the practical importance of identifiable parameters.

## Abstract

Spatial process models popular in geostatistics often represent the observed data as the sum of a smooth underlying process and white noise. The variation in the white noise is attributed to measurement error, or micro-scale variability, and is called the "nugget". We formally establish results on the identifiability and consistency of the nugget in spatial models based upon the Gaussian process within the framework of in-fill asymptotics, i.e. the sample size increases within a sampling domain that is bounded. Our work extends results in fixed domain asymptotics for spatial models without the nugget. More specifically, we establish the identifiability of parameters in the Mat\'ern covariance function and the consistency of their maximum likelihood estimators in the presence of discontinuities due to the nugget. We also present simulation studies to demonstrate the role of the identifiable quantities in spatial interpolation.