In this paper, the varying-coefficient single-index model (VCSIM) is proposed. It can be seen as a generalization of the semivaryingcoefficient model by changing its constant coefficient part to a nonparametric component, or a generalization of the partially linear single-index model by replacing the constant coefficients of its linear part with varying coefficients. Based on the local linear method, average method and backfitting technique, the estimates of the unknown parameters and the unknown functions of the VCSIM are obtained and their asymptotic distributions are derived. Both simulated and real data examples are given to illustrate the model and the proposed estimation methodology.

Varying-coefficient models are popular multivariate nonparametric fitting techniques. When all coefficient functions in a varyingcoefficient model share the same smoothing variable, inference tools available include the F-test, the sieve empirical likelihood ratio test and the generalized likelihood ratio (GLR) test. However, when the coefficient functions have different smoothing variables, these tools cannot be used directly to make inferences on the model because of the differences in the process of estimating the functions. In this paper, the GLR test is extended to models of the latter case by the efficient estimators of these coefficient functions. Under the null hypothesis the new proposed GLR test follows the 2-distribution asymptotically with scale constant and degree of freedom independent of the nuisance parameters, known as Wilks phenomenon. Further, we have derived its asymptotic power which is shown to achieve the optimal rate of convergence for nonparametric hypothesis testing.A simulation study is conducted to evaluate the test procedure empirically.

本文探讨具有不同光滑变量的变系数模型的建模、估计和估计的渐近性。首先，从实际出发建立模型；然后，使用局部线性方法给出模型中未知函数的初始估计，再使用平均方法，给出它们的平均估计；进一步，给出这些平均估计的渐近正态性。两个模拟例子说明这一估计方法是有效的。

In this article, a procedure for estimating the coefficient functions on the functional-coeffcient regression models with different smoothing variables in different coefficient, functions is defined. Firs step, by the local linear technique and the averaged method, the initial estimates of the coefficient functions are given. Second step, based on the initial estimates, the efficient estimates of the coefficient functions are proposed by a one-step back-fitting procedure. The efficient estimators share the same asymptotic normalities as the local linear estimators for the functional-coefficient models with a single smoothing variable in different functions. Two simulated examples show that the procedure is effective.

本文讨论在数据是强相依的情况下函数系数部分线性模型的估计。首先，采用局部线性方法，给出该模型函数是项函数的估计；然后，使用两阶段方法给出系数函数的估计。并且讨论了函数项函数估计的渐近正态性，以及系数函数估计的弱相合性的渐近正态性。模拟研究显示，这些估计是较为理想的。