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Journal of the American Statistical Association Volume 105, 2010 - Issue 490
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Theory and Methods

Simultaneous Confidence Bands for Penalized Spline Estimators

Tatyana Krivobokova Tatyana Krivobokova is Associate Professor, Courant Research Center PEG and Institute for Mathematical Stochastics, Georg August Universität Göttingen, Platz der Göttinger Sieben 3, 37073 Göttingen, Germany . Thomas Kneib is Professor, Mathematics Department, Carl von Ossietzy Universität Oldenburg, 26111 Oldenburg, Germany. Gerda Claeskens is Professor, ORSTAT and Leuven Statistics Research Center, Katholieke Universiteit Leuven, Naamsestraat 69, 3000 Leuven, Belgium. The authors thank the editor, the referees, and Dr. Tatiyana Apanasovich for constructive remarks. They also thank Manuel Wiesenfarth for the substantial improvements to the R program used for the Bayesian confidence bands. The first author acknowledges the support of the German Research Foundation (Deutsche Forschungsgemeinschaft) as part of the Institutional Strategy of the University of Göttingen.
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Thomas Kneib Tatyana Krivobokova is Associate Professor, Courant Research Center PEG and Institute for Mathematical Stochastics, Georg August Universität Göttingen, Platz der Göttinger Sieben 3, 37073 Göttingen, Germany . Thomas Kneib is Professor, Mathematics Department, Carl von Ossietzy Universität Oldenburg, 26111 Oldenburg, Germany. Gerda Claeskens is Professor, ORSTAT and Leuven Statistics Research Center, Katholieke Universiteit Leuven, Naamsestraat 69, 3000 Leuven, Belgium. The authors thank the editor, the referees, and Dr. Tatiyana Apanasovich for constructive remarks. They also thank Manuel Wiesenfarth for the substantial improvements to the R program used for the Bayesian confidence bands. The first author acknowledges the support of the German Research Foundation (Deutsche Forschungsgemeinschaft) as part of the Institutional Strategy of the University of Göttingen.
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Gerda Claeskens Tatyana Krivobokova is Associate Professor, Courant Research Center PEG and Institute for Mathematical Stochastics, Georg August Universität Göttingen, Platz der Göttinger Sieben 3, 37073 Göttingen, Germany . Thomas Kneib is Professor, Mathematics Department, Carl von Ossietzy Universität Oldenburg, 26111 Oldenburg, Germany. Gerda Claeskens is Professor, ORSTAT and Leuven Statistics Research Center, Katholieke Universiteit Leuven, Naamsestraat 69, 3000 Leuven, Belgium. The authors thank the editor, the referees, and Dr. Tatiyana Apanasovich for constructive remarks. They also thank Manuel Wiesenfarth for the substantial improvements to the R program used for the Bayesian confidence bands. The first author acknowledges the support of the German Research Foundation (Deutsche Forschungsgemeinschaft) as part of the Institutional Strategy of the University of Göttingen.
Pages 852-863 | Received 01 Mar 2009, Published online: 01 Jan 2012
 
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Abstract

In this article we construct simultaneous confidence bands for a smooth curve using penalized spline estimators. We consider three types of estimation methods: (a) as a standard (fixed effect) nonparametric model, (b) using the mixed-model framework with the spline coefficients as random effects, and (c) a full Bayesian approach. The volume-of-tube formula is applied for the first two methods and compared with Bayesian simultaneous confidence bands from a frequentist perspective. We show that the mixed-model formulation of penalized splines can help obtain, at least approximately, confidence bands with either Bayesian or frequentist properties. Simulations and data analysis support the proposed methods. The R package ConfBands accompanies the article.

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