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Statistical spatial series modelling II: Some further results on unilateral lattice processes

Published online by Cambridge University Press:  01 July 2016

Dag Tjøstheim
Dag Tjøstheim*
Affiliation:
University of Bergen
*
∗Postal address: Department of Mathematics, University of Bergen, N-5014 Bergen-U, Norway.
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Abstract

An asymptotic theory of estimation is developed for classes of spatial series F(x1, · ··, xn), where (x1, · ··, xn) varies over a regular cartesian lattice. Two classes of unilateral models are studied, namely half-space models and causal (quadrant-type) models. It is shown that a number of asymptotic results are common for these models. Of special interest for practical applications is the problem of determining how many parameters should be included to describe the degree of dependence in each direction. Here we are able to obtain weakly consistent generalizations of familiar time-series criteria under the assumption that the generating variables of the model are independently and identically distributed. For causal models we introduce the concepts of spatial innovation process and lattice martingale and use these to extend some of the asymptotic theory to the case where a certain type of dependence is permitted in the generating variables.

Keywords

LATTICE SERIESUNILATERAL MODELASYMPTOTIC THEORYSPATIAL ORDER DETERMINATIONSPATIAL INNOVATIONLATTICE MARTINGALE
Type
Research Article
Information
Advances in Applied Probability , Volume 15 , Issue 3 , September 1983 , pp. 562 - 584
Copyright
Copyright © Applied Probability Trust 1983 

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