SMOOTHING SPLINE MODELS WITH CORRELATED RANDOM ERRORS

Spline-smoothing techniques are commonly used to estimate the mean fun ction in a nonparametric regression model. Their performances depend g reatly on the choice of smoothing parameters. Many methods of selectin g smoothing parameters such as generalized maximum likelihood (GML), g eneralized 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 thi s article, I assume that observations are correlated and that the corr elation matrix depends on a parsimonious set of parameters. I extend t he GML, GCV, and UBR methods to estimate the smoothing parameters and the correlation parameters simultaneously. I also relate a smoothing s pline model to three mixed-effects models. These relationships show th at 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 restri cted maximum likelihood (REML) estimates. They also provide a way to f it a spline model with correlated errors using the SAS procedure proc mixed. Simulations are conducted to evaluate and compare the performan ce of the GML, GCV, UBR methods and the method proposed by Diggle and Hutchinson. The GML method is recommended, because it is stable and wo rks well in all simulations. It performs better than other methods, es pecially when the sample size is not large. I illustrate my methods wi th applications to time series data and to spatial data.