In this paper, we are concerned with statistical inference for the index parameter α0 in the single-index model Y=g (αT0X) + ε. Based on the estimates obtained by the local linear method, we extend the generalized likelihood ratio test to the single-index model. We investigate the asymptotic behaviour of the proposed test and demonstrate that its limiting null distribution follows a 2-distribution, with the scale constant and the number of degrees of freedom being independent of nuisance parameters or functions, which is called the Wilks phenomenon. A simulated example is used to illustrate the performance of the testing approach.

To study the relationship between the levels of chemical pollutants and the number of daily total hospital admissions for respiratory diseases and to find the effect of temperature/relative humidity on the admission number, Wong et al. [17] introduced the varying-coefficient single-index model (VCSIM). As pointed out, it is a popular multivariate nonparametric fitting technique. However, the tests of the model have not been very well developed. In this paper, based on the estimators obtained by the local linear technique, the average method and the one-step back-fitting technique in the VCSIM, the generalized likelihood ratio (GLR) tests for varying-coefficient parts on the VCSIM are established. Under the null hypotheses the new proposed GLR tests follow the Â2-distribution asymptotically with scale constant and degree of freedom independent of the nuisance parameters, known as Wilks phenomenon. Simulations are conducted to evaluate the test procedure empirically. A real example is used to illustrate the performance of the testing approach.

讨论具有不同自变量的变系数模型的函数系数的估计及其大样本性质使用局部线性方法和积分方法，得到函数系数的积分估计由于该估计有较大的方差，进一步使用回切法改进这一估计，获得了函数系数的改进估计同时，研究了改进估计的渐近正态性最后，用模拟例子说明提出的估计方法是有效的。

This paper considers statistical inference for semiparametric varying coefficient partially linear models with error-prone linear covariates. An empirical likelihood based statistic for parametric component is developed to construct confidence regions. The resulting statistic is shown to be asymptotically chi-square distributed. By the empirical likelihood ratio function, the maximum empirical likelihood estimator of the parameter is defined and the asymptotic normality is shown. A simulation experiment is conducted to compare the empirical likelihood, normal based and the naive empirical likelihood methods in terms of coverage accuracies of confidence regions.

半变系数模型已经获得了广泛的研究和应用，近几年，人们提出许多方法来估计其函数系数和常系数。在PLS方法基础上，本文给出半变系数模型模型在线性随机约束条件下的估计，并证明了常系数和函数系数估计的渐近正态性。