Smoothing Spline Models With Correlated Random Errors

Spline smoothing is a popular method of estimating the functions in a nonparametric regression model. Its performance greatly depends on the choice of smoothing parameters. Many methods of selecting smoothing parameters such as CV, GCV, UBR and GML are developed under the assumption of independent observations. They fail badly when data are correlated. In this paper, we assume observations are correlated and the correlation matrix depends on a parsimonious set of parameters. We extend the GML, GCV and UBR methods to estimate the smoothing parameters and the correlation parameters simultaneously. We also connect a smoothing spline model with three mixed-effects models. These connections show that the smoothing spline estimates evaluated at design points are BLUP estimates and the GML estimates of the smoothing parameters and the correlation parameters are REML estimates. These connections also suggest a way to fit a spline model with correlated errors using the existing SAS procedure pr..