Free access to Scandinavian Journal of Statistics article for limited period


  • Author: Giampiero Massa, Simon N. Wood and Statistics Views
  • Date: 09 July 2013

Each week, we select a new article hot off the press with free access for a limited period. This week's is from Scandinavian Journal of Statistics.

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Giampiero Marra and Simon N. Wood have studied the coverage properties of Bayesian confidence intervals for the smooth component functions of generalized additive models (GAMs) represented using any penalized regression spline approach. The intervals are the usual generalization of the intervals first proposed by Wahba and Silverman in 1983 and 1985, respectively, to the GAM component context. We present simulation evidence showing these intervals have close to nominal ‘across-the-function’ frequentist coverage probabilities, except when the truth is close to a straight line/plane function. We extend the argument introduced by Nychka in 1988 for univariate smoothing splines to explain these results.

The theoretical argument suggests that close to nominal coverage probabilities can be achieved, provided that heavy oversmoothing is avoided, so that the bias is not too large a proportion of the sampling variability. The theoretical results allow us to derive alternative intervals from a purely frequentist point of view, and to explain the impact that the neglect of smoothing parameter variability has on confidence interval performance. They also suggest switching the target of inference for component-wise intervals away from smooth components in the space of the GAM identifiability constraints.

Massa and Woods find that, by simulation and extension of Nychka's (1988) analysis the Wahba/Silverman type Bayesian intervals for the components of a penalized regression spline based GAM have generally close to nominal frequentist properties, across-the-function. As simulation evidence and our theoretical arguments suggest, the exception occurs when components estimated subject to identifiability constraints have interval widths vanishing somewhere as a result of heavy smoothing. Coverage probabilities can be improved if intervals are only obtained for unconstrained quantities, such as a smooth component plus the model intercept. The theoretical results also allow us to define alternative intervals from a purely frequentist approach, and these appear to perform almost as well as the Bayesian intervals.

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