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Biometrika 2008 95(2):415-436; doi:10.1093/biomet/asn010
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© 2008 Biometrika Trust

Articles

On the asymptotics of penalized splines

Yingxing Li

Department of Statistical Science, Malott Hall, Cornell University, New York 14853, U.S.A. yl377{at}cornell.edu

David Ruppert

School of Operational Research and Information Engineering, Rhodes Hall, Cornell University, New York 14853, U.S.A. dr24{at}cornell.edu

Received for publication 1 September 2006. Revision received 1 September 2007.
  Abstract

We study the asymptotic behaviour of penalized spline estimators in the univariate case. We use B-splines and a penalty is placed on mth-order differences of the coefficients. The number of knots is assumed to converge to infinity as the sample size increases. We show that penalized splines behave similarly to Nadaraya--Watson kernel estimators with ‘equivalent’ kernels depending upon m. The equivalent kernels we obtain for penalized splines are the same as those found by Silverman for smoothing splines. The asymptotic distribution of the penalized spline estimator is Gaussian and we give simple expressions for the asymptotic mean and variance. Provided that it is fast enough, the rate at which the number of knots converges to infinity does not affect the asymptotic distribution. The optimal rate of convergence of the penalty parameter is given. Penalized splines are not design-adaptive.

Key Words: Asymptotic bias • Binning • B-spline • Difference penalty • Equivalent kernel • Increasing number of knots • P-spline


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