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Biometrika 2004 91(1):177-193; doi:10.1093/biomet/91.1.177
© 2004 by Biometrika Trust
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Equivalent kernels of smoothing splines in nonparametric regression for clustered/longitudinal data

Xihong Lin1, Naisyin Wang2, Alan H.Welsh3 and Raymond J.Carroll4

1 Department of Biostatistics, University of Michigan, Ann Arbor, Michigan 48109, U.S.Axlin{at}umich.edu 2 Department of Statistics, Texas A&M University, College Station, Texas 77843-3143, U.S.A.nwang{at}stat.tamu.edu 3 Faculty of Mathematical Studies, University of Southampton, Southampton, Hampshire SO17 1BJ, U.K. a.h.welsh{at}maths.soton.ac.uk 4 Department of Statistics, Texas A&M University, College Station, Texas 77843-3143, U.S.A. carroll{at}stat.tamu.edu

For independent data, it is well known that kernel methods and spline methods are essentially asymptotically equivalent (Silverman, 1984).However, recent work of Welsh et al. (2002) shows that the same is not true for clustered/longitudinal data. Splines and conventional kernels are different in localness and ability to account for the within-cluster correlation. We show that a smoothing spline estimator is asymptotically equivalent to a recently proposed seemingly unrelated kernel estimator of Wang (2003) for any working covariance matrix. We show that both estimators can be obtained iteratively by applying conventional kernel or spline smoothing to pseudo-observations. This result allows us to study the asymptotic properties of the smoothing spline estimator by deriving its asymptotic bias and variance. We show that smoothing splines are consistent for an arbitrary working covariance and have the smallest variance when assuming the true covariance. We further show that both the seemingly unrelated kernel estimator and the smoothing spline estimator are nonlocal unless working independence is assumed but have asymptotically negligible bias. Their finite sample performance is compared through simulations. Our results justify the use of efficient, non-local estimators such as smoothing splines for clustered/longitudinal data.

Key Words: Asymptotic bias and variance; Asymptotic equivalent kernel; Consistency; Kernel regression; Longitudinal data; Non-localness; Nonparametric regression; Smoothing spline regression

Received December 2002. Revised July 2003


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