Articles |
On the asymptotics of penalized splines
Department of Statistical Science, Malott Hall, Cornell University, New York 14853, U.S.A. yl377{at}cornell.edu
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.
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
References
-
Agarwal G. G., Studden W. J. Asymptotic integrated mean square error using least squares and bias minimizing splines. Ann. Statist. (1980) 8:1307–25.[CrossRef]
Dow M. Explicit inverses of Toeplitz and associated matrices. ANZIAM J. (2003) 44:E185–E215.
Eilers P. H. C., Marx B. D. Flexible smoothing with B-splines and penalties (with Discussion). Statist. Sci. (1996) 11:89–121.[CrossRef]
Elaydi S. An Introduction to Difference Equations (2005) New York: Springer.
Fan J. Design-adaptive nonparametric regression. J. Am. Statist. Assoc. (1992) 87:998–1004.[CrossRef][Web of Science]
Gradshteyn I., Ryzhik I. Table of Integrals, Series, and Products. (1980) New York: Academic Press.
Hall P., Opsomer J. D. Theory for penalized spline regression. Biometrika (2005) 92:105–18.
Hall P., Wand M. P. On the accuracy of binned kernel density estimators. J. Mult. Anal. (1996) 5:165–84.
O'Sullivan F. A statistical perspective on ill-posed inverse problems (with Discussion). Statist. Sci. (1986) 1:505–27.
Ruppert D. Selecting the number of knots for penalized splines. J. Comp. Graph. Statist. (2002) 11:735–57.[CrossRef]
Ruppert D., Wand M. P. Multivariate locally weighted least squares regression. In: Ann. Statist. (1994) 22:1346–69.[CrossRef]
Ruppert D., Wand M. P., Carroll R. J. Semiparametric Regression (2003) Cambridge: Cambridge University Press.
Silverman B. W. Spline smoothing: the equivalent variable kernel method. Ann. Statist. (1984) 12:898–916.[CrossRef]
Stute W. The oscillation behavior of empirical processes. Ann. Prob. (1982) 10:86–107.[CrossRef]
Stute W. Asymptotic normality of nearest neighbor regression function estimates. Ann. Statist. (1984) 12:917–26.[CrossRef]
Wand M. P. On the optimal amount of smoothing in penalized spline regression. Biometrika (1999) 86:936–40.
Wand M. P., Jones M. C. Kernel Smoothing (1995) London: Chapman & Hall/CRC.
Yu Y., Ruppert D. Penalized spline estimation for partially linear single index model. J. Am. Statist. Assoc. (2002) 97:1042–54.[CrossRef][Web of Science]
CiteULike 
Connotea 
Del.icio.us What's this?
This article has been cited by other articles:
|
|
 |
|
  G. Claeskens, T. Krivobokova, and J. D. Opsomer Asymptotic properties of penalized spline estimators Biometrika, September 1, 2009; 96(3): 529 - 544. [Abstract] [PDF]
|
|
| ||||||||||||||||||||||||||||||||||||||||||||||||||