A goodness-of-fit test for regression models with spatially correlated errors
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A goodness-of-fit test for regression models with spatially correlated errors Andrea Meilán-Vila1 Rosa M. Crujeiras3
· Jean D. Opsomer2 · Mario Francisco-Fernández1
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Received: 10 January 2019 / Accepted: 9 September 2019 © Sociedad de Estadística e Investigación Operativa 2019
Abstract The problem of assessing a parametric regression model in the presence of spatial correlation is addressed in this work. For that purpose, a goodness-of-fit test based on a L 2 -distance comparing a parametric and nonparametric regression estimators is proposed. Asymptotic properties of the test statistic, both under the null hypothesis and under local alternatives, are derived. Additionally, a bootstrap procedure is designed to calibrate the test in practice. Finite sample performance of the test is analyzed through a simulation study, and its applicability is illustrated using a real data example. Keywords Model checking · Spatial correlation · Local linear regression · Least squares · Bootstrap Mathematics Subject Classification 62G10 · 62H11 · 62G08 · 62G09
1 Introduction The problem of testing a parametric regression model, confronting a parametric estimator of the regression function with a smooth alternative estimated by a nonparametric method, has been approached by several authors in the statistical literature
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s11749019-00678-y) contains supplementary material, which is available to authorized users.
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Andrea Meilán-Vila [email protected]
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Research group MODES, Department of Mathematics, Faculty of Computer Science, Universidade da Coruña, Campus de Elviña s/n, 15071 A Coruña, Spain
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Westat, 1600 Research Boulevard, Rockville, MD 20850, USA
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Research group MODESTYA, Department of Statistics, Mathematical Analysis and Optimization, Faculty of Mathematics, Universidade de Santiago de Compostela, Rúa Lope Gómez de Marzoa s/n, 15782 Santiago de Compostela, Spain
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(see, for example Azzalini et al. 1989; Eubank and Spiegelman 1990). For instance, Weihrather (1993) and Eubank et al. (2005) described tests based on an overall distance between parametric and nonparametric regression fits, giving some strategies on bandwidth selection. Härdle and Mammen (1993) proposed a testing procedure to check if a regression function belongs to a class of parametric models by measuring a L 2 -distance between parametric and nonparametric estimates. Specifically, the Nadaraya–Watson estimator (Nadaraya 1964; Watson 1964) was considered for the nonparametric approach. The same type of study was performed by Alcalá et al. (1999), but using a local polynomial regression estimator (Fan and Gijbels 1996). Following similar ideas, a local test for a univariate parametric model checking was proposed by Opsomer and Francisco-Fernández (2010), while Li (2005) assessed the lack of fit of a nonlinear regression model, comparing a local linear smoother and parametric fits. The previous testing procedures, all of them
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