Smoothing Spline Models with Correlated Random Errors

Abstract

Spline-smoothing techniques are commonly used to estimate the mean function in a nonparametric regression model. Their performances depend greatly on the choice of smoothing parameters. Many methods of selecting smoothing parameters such as generalized maximum likelihood (GML), generalized cross-validation (GCV), and unbiased risk (UBR), have been developed under the assumption of independent observations. They tend to underestimate smoothing parameters when data are correlated. In this article, I assume that observations are correlated and that the correlation matrix depends on a parsimonious set of parameters. I extend the GML, GCV, and UBR methods to estimate the smoothing parameters and the correlation parameters simultaneously. I also relate a smoothing spline model to three mixed-effects models. These relationships show that the smoothing spline estimates evaluated at design points are best linear unbiased prediction (BLUP) estimates and that the GML estimates of the smoothing parameters and the correlation parameters are restricted maximum likelihood (REML) estimates. They also provide a way to fit a spline model with correlated errors using the SAS procedure proc mixed. Simulations are conducted to evaluate and compare the performance of the GML, GCV, UBR methods and the method proposed by Diggle and Hutchinson. The GML method is recommended, because it is stable and works well in all simulations. It performs better than other methods, especially when the sample size is not large. I illustrate my methods with applications to time series data and to spatial data.