Global and local diagnostic analytics for a geostatistical model based on a new approach to quantile regression
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ORIGINAL PAPER
Global and local diagnostic analytics for a geostatistical model based on a new approach to quantile regression Vı´ctor Leiva1
•
Luis Sa´nchez2
•
Manuel Galea3
•
Helton Saulo4
Ó Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Data with spatial dependence are often modeled by geoestatistical tools. In spatial regression, the mean response is described using explanatory variables with georeferenced data. This modeling frequently considers Gaussianity assuming the response follows a symmetric distribution. However, when this assumption is not satisfied, it is useful to suppose distributions with the same asymmetric behavior of the data. This is the case of the Birnbaum–Saunders (BS) distribution, which has been considered in different areas and particularly in environmental sciences due to its theoretical arguments. We propose a geostatistical model based on a new approach to quantile regression considering the BS distribution. Global and local diagnostic analytics are derived for this model. The estimation of model parameters and its local influence are conducted by the maximum likelihood method. Global influence is based on the Cook distance and it is compared to local influence, in both cases to detect influential observations, whose detection and removal can modify the conclusions of a study. We illustrate the proposed methodology applying it to environmental data, which shows this situation changing the conclusions after removing potentially influential observations. A comparison with Gaussian spatial regression is conducted. Keywords Diagnostic techniques Environmental data Maximum likelihood method R software Spatial models
1 Introduction The Birnbaum–Saunders (BS) distribution constantly arises in the applied statistical literature. In the last decades, it has been shown to be versatile and efficient in several fields of science, being widely studied, due to its theoretical justification, its good properties and its close relation with the Gaussian or normal model. The BS distribution is unimodal, with asymmetry to the right and support defined on & Vı´ctor Leiva [email protected] https://www.victor.leiva.cl 1
School of Industrial Engineering, Pontificia Universidad Cato´lica de Valparaı´so, Valparaı´so, Chile
2
Department of Mathematics and Statistics, Universidad de La Frontera, Temuco, Chile
3
Department of Statistics, Pontificia Universidad Cato´lica de Chile, Santiago, Chile
4
Department of Statistics, Universidade de Brası´lia, Brası´lia, Brazil
the positive real numbers, in addition to being indexed by two parameters that control its shape and scale. The BS distribution has its origins in physics and engineering, being it derived to model a fatigue phenomenon related to crack development in metallic objects; see Leiva (2019). Recently, Budsaba et al. (2020) suggested a physical model of this phenomenon which shows exactly how Birnbaum and Saunders (1969) derived their distribution to fit the model of a crack development. However, at
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