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On the smallest eigenvalues of covariance matrices of multivariate spatial processes
Author(s) -
Bachoc François,
Furrer Reinhard
Publication year - 2016
Publication title -
stat
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.61
H-Index - 18
ISSN - 2049-1573
DOI - 10.1002/sta4.107
Subject(s) - matérn covariance function , mathematics , rational quadratic covariance function , covariance function , covariance , covariance matrix , univariate , estimator , estimation of covariance matrices , multivariate statistics , scatter matrix , law of total covariance , eigenvalues and eigenvectors , parametric statistics , bounded function , statistics , mathematical analysis , covariance intersection , physics , quantum mechanics
There has been a growing interest in providing models for multivariate spatial processes. A majority of these models specify a parametric matrix covariance function. Based on observations, the parameters are estimated by maximum likelihood or variants thereof. While the asymptotic properties of maximum likelihood estimators for univariate spatial processes have been analyzed in detail, maximum likelihood estimators for multivariate spatial processes have not received their deserved attention yet. In this article, we consider the classical increasing‐domain asymptotic setting restricting the minimum distance between the locations. Then, one of the main components to be studied from a theoretical point of view is the asymptotic positive definiteness of the underlying covariance matrix. Based on very weak assumptions on the matrix covariance function, we show that the smallest eigenvalue of the covariance matrix is asymptotically bounded away from zero. Several practical implications are discussed as well. Copyright © 2016 John Wiley & Sons, Ltd.