Abstract
The validity of estimation and smoothing parameter selection for the wide class of generalized additive models for location, scale and shape (GAMLSS) relies on the correct specification of a likelihood function. Deviations from such assumption are known to mislead any likelihood-based inference and can hinder penalization schemes meant to ensure some degree of smoothness for nonlinear effects. We propose a general approach to achieve robustness in fitting GAMLSSs by limiting the contribution of observations with low log-likelihood values. Robust selection of the smoothing parameters can be carried out either by minimizing information criteria that naturally arise from the robustified likelihood or via an extended Fellner–Schall method. The latter allows for automatic smoothing parameter selection and is particularly advantageous in applications with multiple smoothing parameters. We also address the challenge of tuning robust estimators for models with nonlinear effects by proposing a novel median downweighting proportion criterion. This enables a fair comparison with existing robust estimators for the special case of generalized additive models, where our estimator competes favorably. The overall good performance of our proposal is illustrated by further simulations in the GAMLSS setting and by an application to functional magnetic resonance brain imaging using bivariate smoothing splines.
Highlights
Generalized additive models for location, scale and shape (GAMLSS) are flexible nonparametric regression models that have been introduced by Rigby and Stasinopoulos (2005); see the recent book and tutorial by Stasinopoulos et al (2017) and Stasinopoulos et al (2018) for a review
For the selection of the smoothing parameters, we propose robust versions of the Akaike information criterion (AIC) and Bayesian information criterion (BIC), that can be typically minimized in a grid search, and an adaptation of the Fellner–Schall automatic multiple smoothing parameter selection method (Wood and Fasiolo 2017), which has important practical advantages
Since we only have one smooth term here, we can afford the computational cost of the brute-force CV of AS and consider three variants of our estimator to compare smoothing parameter selection methods: minimizing our proposed robust AIC (RAIC); minimizing our robust BIC (RBIC); and the extended Fellner–Schall method (EFS)
Summary
Generalized additive models for location, scale and shape (GAMLSS) are flexible nonparametric regression models that have been introduced by Rigby and Stasinopoulos (2005); see the recent book and tutorial by Stasinopoulos et al (2017) and Stasinopoulos et al (2018) for a review.
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