Inference and diagnostics in spatial models

Abstract

In this work, we present inference and diagnostics in spatial models. Firstly, we extend the Gaussian spatial linear model for the elliptical spatial linear models, and present the local influence methodology to assess the sensitivity of the maximum likelihood estimators to small perturbations in the data and/or the spatial linear model assumptions. Secondly, we consider the Gaussian spatial linear models with repetitions. We obtain in matrix notation a Bartlett correction factor for the profiled likelihood ratio statistic. We also present inference approach to estimate the smooth parameter from the Mat´ern family class of models. The maximum likelihood estimators are obtained, and an explicit expression for the Fisher information matrix is also presented, even when the smooth parameter for Mat´ern class of covariance structure is estimated. We present local and global influence diagnostics techniques to assess the influence of observations on Gaussian spatial linear models with repetitions. We review concepts of Cook’s distance and generalized leverage and extend it. For local influence we consider two different approach and for both we consider appropriated perturbation in the response variable and case weight perturbation. Finally, we describe the modeling and fitting of Markov random field spatial components within the generalized additive models for locations scale and shape framework. This allows modeling any or all of the parameters of the distribution for the response variable using explanatory variables and spatial effects. We present some simulations and real data sets illustrate the methodology.