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Asymptotic distributions of M-estimators in a spatial regression model under some fixed and stochastic spatial sampling design

Research output: Contribution to journalJournal article

Published

Journal publication date2004
JournalAnnals of the Institute of Statistical Mathematics
Journal number2
Volume56
Number of pages26
Pages225-250
Original languageEnglish

Abstract

In this paper, we consider M-estimators of the regression parameter in a spatial multiple linear regression model. We establish consistency and asymptotic normality of the M-estimators when the data-sites are generated by a class of deterministic as well as a class of stochastic spatial sampling schemes. Under the deterministic sampling schemes, the data-sites are located on a regular grid but may have an infill component. On the other hand, under the stochastic sampling schemes, locations of the data-sites are given by the realizations of a collection of independent random vectors and thus, are irregularly spaced. It is shown that scaling constants of different orders are needed for asymptotic normality under different spatial sampling schemes considered here. Further, in the stochastic case, the asymptotic covariance matrix is shown to depend on the spatial sampling density associated with the stochastic design. Results are established for M-estimators corresponding to certain non-smooth score functions including Huber's e-function and the sign functions (corresponding to the sample quantiles).