Smoothing parameter and model selection for general smooth models (with discussion)

Simon N Wood, Natalya Pya, Benjamin Saefken

Research output: Contribution to journalArticle (Academic Journal)peer-review

327 Citations (Scopus)
1228 Downloads (Pure)


This paper discusses a general framework for smoothing parameter estimation for models with regular likelihoods constructed in terms of unknown smooth functions of covariates. Gaussian random effects and parametric terms may also be present. By construction the method is numerically stable and convergent, and enables smoothing parameter uncertainty to be quantified. The latter enables us to fix a well known problem with AIC for such models, thereby improving the range of model selection tools available. The smooth functions are represented by reduced rank spline like smoothers, with associated quadratic penalties measuring function smoothness. Model estimation is by penalized likelihood maximization, where the smoothing parameters controlling the extent of penalization are estimated by Laplace approximate marginal likelihood. The methods cover, for example, generalized additive models for non-exponential family responses (for example beta, ordered categorical, scaled t distribution, negative binomial and Tweedie distributions), generalized additive models for location scale and shape (for example two stage zero inflation models, and Gaussian location-scale models), Cox proportional hazards models and multivariate additive models. The framework reduces the implementation of new model classes to the coding of some standard derivatives of the log likelihood.
Original languageEnglish
Pages (from-to)1548-1563
Number of pages28
JournalJournal of the American Statistical Association
Issue number516
Publication statusPublished - 4 Jan 2017

Bibliographical note

E-published ahead of print: 4 January 2017

Structured keywords

  • Jean Golding


  • Additive model
  • AIC
  • Distributional regression
  • GAM
  • Location scale and shape model
  • Ordered categorical regression
  • Penalized regression spline
  • REML
  • Smooth Cox model
  • Smoothing parameter uncertainty
  • Statistical algorithm
  • Tweedie distribution


Dive into the research topics of 'Smoothing parameter and model selection for general smooth models (with discussion)'. Together they form a unique fingerprint.

Cite this