# STATISTICAL ESTIMATION OFVARIOGRAM AND COVARIANCEPARAMETERS OF SPATIAL ANDSPATIO-TEMPORAL RANDOMPROCESES

• Sourav Das

Student thesis: Phd

### Abstract

In this thesis we study the problem of estimation of parametric covariance and variogramfunctions for spatial and spatio- temporal random processes. It has the followingprincipal parts.Variogram EstimationWe consider the "weighted" least squares criterion of fitting a parametric variogramfunction to second order stationary geo-statistical processes. Two new weight functionsare investigated as alternative to the commonly used weight function proposedby Cressie (1985). We discuss asymptotic convergence properties of the sample variogramestimator and estimators of unknown parameters of parametric variogramfunctions, under a "mixed increasing domain" sampling design as proposed by Lahiriet al.. While empirical results of Mean Square Errors, for parameter estimation, obtainedusing both the proposed functions are found to be comparatively better, wealso theoretically establish that under general conditions one of the proposed weightfunctions give estimates with better asymptotic effciency.Spatio-Temporal Covariance EstimationOver the past decade, there have been some important advances in methods for constructingvalid spatiotemporal covariance functions; but not much attention has beengiven -so far- on methods of parameter estimation. In this thesis we propose a newfrequency domain approach to estimating parameters of spatio-temporal covariancefunctions. We derive asymptotic strong consistency properties of the estimators usingthe concept of stochastic equicontinuity. The theory is illustrated with a simulation.Non-Linearity of Geostatistical DataLinear prediction theory for spatial data is well established and substantial literatureis available on the subject. Relatively less is known about non-linearity. In our finaland ongoing, research problem we propose a non-linear predictor for geostatisticaldata. We demonstrate that the predictor is a function of higher order moments. Thisleads us to construct spatial bispectra for parametric third order moments.
Date of Award 31 Dec 2011 English The University of Manchester Georgi Boshnakov (Supervisor)

### Keywords

• Whittle Likelihood
• Variogram Estimation

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