The Annals of Statistics

Maximum Likelihood Estimation of Parameters under a Spatial Sampling Scheme

Zhiliang Ying

Full-text: Open access


We study in detail asymptotic properties of maximum likelihood estimators of parameters when observations are taken from a two-dimensional Gaussian random field with a multiplicative Ornstein-Uhlenbeck covariance function. Under the complete lattice sampling plan, it is shown that the maximum likelihood estimators are strongly consistent and asymptotically normal. The asymptotic normality here is normalized by the fourth root of the sample size and is obtained through higher order expansions of the likelihood score equations. Extensions of these results to higher-dimensional processes are also obtained, showing that the convergence rate becomes better as the dimension gets higher.

Article information

Ann. Statist., Volume 21, Number 3 (1993), 1567-1590.

First available in Project Euclid: 12 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62F12: Asymptotic properties of estimators
Secondary: 60G60: Random fields 60G15: Gaussian processes

Gaussian random fields multiplicative covariance functions lattice sampling maximum likelihood estimation computer experiments consistency asymptotic normality


Ying, Zhiliang. Maximum Likelihood Estimation of Parameters under a Spatial Sampling Scheme. Ann. Statist. 21 (1993), no. 3, 1567--1590. doi:10.1214/aos/1176349272.

Export citation