Robust and efficient estimation of nonparametric generalized linear models

Generalized linear models are flexible tools for the analysis of diverse datasets, but the classical formulation requires that the parametric component is correctly specified and the data contain no atypical observations. To address these shortcomings, we introduce and study a family of nonparametric full-rank and lower-rank spline estimators that result from the minimization of a penalized density power divergence. The proposed class of estimators is easily implementable, offers high protection against outlying observations and can be tuned for arbitrarily high efficiency in the case of clean data. We show that under weak assumptions, these estimators converge at a fast rate and illustrate their highly competitive performance on a simulation study and two real-data examples. SUPPLEMENTARY INFORMATION: The online version contains supplementary material available at 10.1007/s11749-023-00866-x.