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Biometrika Advance Access published online on October 29, 2008
Biometrika, doi:10.1093/biomet/asn041
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© 2008 Biometrika Trust

Article

On the asymptotics of marginal regression splines with longitudinal data

Zhongyi Zhu

Department of Statistics, Fudan University, Shanghai 200433, China zhuzy{at}fudan.edu.cn

Wing K. Fung

Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong, China wingfung{at}hku.hk

Xuming He

Department of Statistics, University of Illinois at Urbana-Champaign, 725 South Wright Street, Champaign, Illinois 61820, U.S.A. x-he{at}uiuc.edu

Received for publication 1 March 2007. Revision received 1 April 2008.
  Abstract

There have been studies on how the asymptotic efficiency of a nonparametric function estimator depends on the handling of the within-cluster correlation when nonparametric regression models are used on longitudinal or cluster data. In particular, methods based on smoothing splines and local polynomial kernels exhibit different behaviour. We show that the generalized estimation equations based on weighted least squares regression splines for the nonparametric function have an interesting property: the asymptotic bias of the estimator does not depend on the working correlation matrix, but the asymptotic variance, and therefore the mean squared error, is minimized when the true correlation structure is specified. This property of the asymptotic bias distinguishes regression splines from smoothing splines.

Key Words: Asymptotic bias • B-spline • Generalized estimating equation • Generalized linear model • Least squares • Longitudinal data.


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