Smoothing Parameter and Model Selection for General Smooth Models

Abstract

This article discusses a general framework for smoothing parameter estimation for models with regular likelihoods
\nconstructed in terms of unknown smooth functions of covariates. Gaussian random effects and
\nparametric terms may also be present. By construction the method is numerically stable and convergent,
\nand enables smoothing parameter uncertainty to be quantified. The latter enables us to fix a well known
\nproblem with AIC for such models, thereby improving the range of model selection tools available. The
\nsmooth functions are represented by reduced rank spline like smoothers, with associated quadratic penalties
\nmeasuring function smoothness. Model estimation is by penalized likelihood maximization, where
\nthe smoothing parameters controlling the extent of penalization are estimated by Laplace approximate
\nmarginal likelihood. The methods cover, for example, generalized additive models for nonexponential family
\nresponses (e.g., beta, ordered categorical, scaled t distribution, negative binomial and Tweedie distributions),
\ngeneralized additive models for location scale and shape (e.g., two stage zero inflation models, and
\nGaussian location-scale models), Cox proportional hazards models and multivariate additive models. The
\nframework reduces the implementation of new model classes to the coding of some standard derivatives
\nof the log-likelihood. Supplementary materials for this article are available online.