B Spline For Varying-Coefficient Single-Index Model

Varying-coefficient single-index model(VCSIM) can be seen as a generalization of the semivarying-coefficient model or partially linear single-index model. It covers a variety of situations, such as single-index model, partially linear single-index model, varying coefficient model, partially linear varying coefficient model. Recently the main research for varying-coefficient single-index model is conducted by using local linear estimation method. The drawback is that it is hard to choose bandwidths when using local linear estimation which is a local estimation method. And we have to estimate the parametric and nonparametric parts separately. Penalized spline and B spline estimation, which are global estimation methods, can directly estimate the parametric and nonparametric parts simultaneously. But penalized spline requires more knots, which means more parameters need to be estimated. It is a challenge for nonlinear least square estimation.In this paper, we will estimate the varying-coefficient single-index model by B spline technique. Two methods for estimation and discussed, the first one is we directly estimate the parametric and nonparametric parts simultaneously by nonlinear least square estima-tion, the second one is to estimate the two parts individually. We suggest that the second method is for our preference when the large amount of parameters are involved, otherwise the first method will be more convenient. Besides, both estimations'asymptotic distri-bution are proved. At last simulated examples are given to illustrate the performance of the model and the proposed estimation methodologies.