On the asymptotics of penalized splines

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 s...

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Bibliographic Details
Published in:Biometrika 2008-06, Vol.95 (2), p.415-436
Main Authors: Li, Yingxing, Ruppert, David
Format: Article
Language:English
Subjects:
Applications
Approximations and expansions
Asymptotic bias
Asymptotic methods
Asymptotic theory
B-spline
Bias
Binning
Biology, psychology, social sciences
Data smoothing
Degrees of polynomials
Difference penalty
Equivalent kernel
Estimation bias
Estimators
Exact sciences and technology
Fines & penalties
General topics
Increasing number of knots
Integers
Knots
Mathematical analysis
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Numerical approximation
P-spline
Polynomials
Probability and statistics
Sample size
Sciences and techniques of general use
Statistics
Studies
Wands
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Description
Summary: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.
ISSN:0006-3444
1464-3510
DOI:10.1093/biomet/asn010