  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
non-linear 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 non-linear
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.

Methodology