The Performance of Cross-Validation and Maximum Likelihood Estimators of Spline Smoothing Parameters

Abstract An important aspect of nonparametric regression by spline smoothing is the estimation of the smoothing parameter. In this article we report on an extensive simulation study that investigates the finite-sample performance of generalized cross-validation, cross-validation, and marginal likelihood estimators of the smoothing parameter in splines of orders 2 and 3. The performance criterion for both the estimate of the function and its first derivative is measured by the square root of integrated squared error. Marginal likelihood using splines of degree 5 emerges as an attractive alternative to the other estimators in that it usually outperforms them and is also faster to compute.

[1]  G. Wahba.  Smoothing noisy data with spline functions , 1975 .

[2]  G. Wahba,et al.  A completely automatic french curve: fitting spline functions by cross validation , 1975 .

[3]  Michael L. Stein,et al.  A Comparison of Generalized Cross Validation and Modified Maximum Likelihood for Estimating the Parameters of a Stochastic Process , 1990 .

[4]  M. C. Jones,et al.  Spline Smoothing and Nonparametric Regression. , 1989 .

[5]  William H. Press,et al.  Numerical recipes in C. The art of scientific computing , 1987 .

[6]  Herman J. Woltring,et al.  A fortran package for generalized, cross-validatory spline smoothing and differentiation , 1986 .

[7]  M. F. Hutchinson,et al.  Algorithm 642: A fast procedure for calculating minimum cross-validation cubic smoothing splines , 1986, TOMS.

[8]  R. Kohn,et al.  A new algorithm for spline smoothing based on smoothing a stochastic process , 1987 .

[9]  Peter Craven,et al.  Smoothing noisy data with spline functions , 1978 .

[10]  R. Kohn,et al.  Estimation, Filtering, and Smoothing in State Space Models with Incompletely Specified Initial Conditions , 1985 .

[11]  G. Wahba.  A Comparison of GCV and GML for Choosing the Smoothing Parameter in the Generalized Spline Smoothing Problem , 1985 .

[12]  R. Kohn,et al.  A fast algorithm for signal extraction, influence and cross-validation in state space models , 1989 .

[13]  B. Silverman,et al.  Spline Smoothing: The Equivalent Variable Kernel Method , 1984 .

[14]  William H. Press,et al.  Numerical recipes , 1990 .

[15]  G. Wahba.  Improper Priors, Spline Smoothing and the Problem of Guarding Against Model Errors in Regression , 1978 .

[16]  Bernard W. Silverman,et al.  A Fast and Efficient Cross-Validation Method for Smoothing Parameter Choice in Spline Regression , 1984 .

[17]  C. Ansley,et al.  The Signal Extraction Approach to Nonlinear Regression and Spline Smoothing , 1983 .

[18]  T. Gasser,et al.  Residual variance and residual pattern in nonlinear regression , 1986 .