Spline Estimator in Multi-Response Nonparametric Regression Model

In many applications two or more dependent variables are observed at several values of the independent variables, such as at time points. The statistical problems are to estimate functions that model their dependences on the independent variables, and to investigate relationships between these functions. Nonparametric regression model, especially smoothing splines provides powerful tools to model the functions which draw association of these variables. Penalized weighted least-squares is used to jointly estimate nonparametric functions from contemporaneously correlated data. In this paper we formulate the multi-response nonparametric regression model and give a theoretical method for both obtaining distribution of the response and estimating the nonparametric function in the model. We also estimate the smoothing parameters, the weighting parameters and the correlation parameter simultaneously by applying three methods: generalized maximum likelihood (GML), generalized cross validation (GCV) and leaving-out-one-pair cross validation (CV).