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
