FITTING SMOOTHING SPLINES TO DATA FROM MULTIPLE SOURCES

The problem of using the ''direct'' variational methods in a statistical model that merges data from different sources with unknown relative weights is considered. To carry out this merging optimally, it is necessary to provide an estimate of the relative weights to be given to data from different sources. A new form of generalized cross validation (GCV) estimate for simultaneously estimating the weighting parameters and the smoothing parameters is developed here. We name this estimate GCV-r, where r represents the weighting parameter. We study the properties of the GCV-r estimators as well as the properties of the generalized maximum likelihood (GML-r) estimators proposed in Wahba, Johnson and Reames (1990). We prove the weak consistency and the asymptotic normality of all these estimators under a stochastic model. The convergence rates for these estimators are obtained under some conditions. Some simulation studies are carried out both to confirm the theoretical results and to compare different methods under different situations. The simulation results show good agreement with the theoretical results. The simulation results also show that the GCV-r is better than the GML-r in estimating the underlying smooth function when the model is misspecified