Spline Smoothing With Correlated Random Errors

Spline smoothing provides a powerful tool for estimating nonparametric functions. Most of the past work is based on the assumption that the random errors are independent. Observations are often correlated in applications; e.g., time series data, spatial data and clustered data. It is well known that correlation greatly a ects the selection of smoothing parameters, which are critical to the performance of smoothing spline estimates. Popular methods for selecting smoothing parameters such as generalized maximum likelihood (GML), generalized cross-validation (GCV) and unbiased risk (UBR) underestimate smoothing parameters when data are positively correlated and the correlation is ignored. This is because that all automatic smoothing parameter selection methods perceive all the trend (signal) in the data is due to the mean function f , and attempt to incorporate that trend into estimate. Correlated random errors may induce \local" trend and thus fools these methods. The model is essentially unidenti able if no parametric shape is assumed for the mean nor the correlation function. We assume the form of the covariance matrix is known. We present extensions of the GML, UBR and GCV method to estimate the smoothing parameter and the correlation parameters simultaneously.