Estimating And Checking For Semiparametric Regression Models With Indexes

The semiparametric regression models with indexes are a kind of important statis-tical models in multivariate nonparametric and semiparametric regression. They mainly include the single-index model, the partially linear single-index model and the varying-coefficient single-index model. By reducing the dimensionality from multivariate predictors to a univariate index, this kind of models avoid the so-called "curse of dimensionality" while still capturing important features in high-dimensionality data. Thus, statistical in-ference for this kind of models is a very important problem in multivariate nonparametric regression, and presently it is a hot topic. This forms the main theme of the study.Firstly we are concerned with statistical inference for the index parameter ao in the single-index model, which is written as where Y (?) R is a response variable, X= (X\1,…,Xq)T (?) Rq are covariates, the un-known index parameter vectorα0= (a01,…,α0q)T is in Rq and‖α0‖= 1 for model identifiability, ao(·) is an unknown univariable measurable function, which is called as index function, the errorεis independent of X with E(ε)= 0 and Var(ε)=σ2. The con-ventional maximum likelihood ratio (MLR) test may not exist in a semiparametric versus another semiparametric regression setting. This is because the nonparametric MLE for the unknown function ao(·) does not exist. Even if the MLE exists, the corresponding MLR test is not optimal. To end this, Fan et al. (2001) proposed a family of test called generalized likelihood ratio test, which is written as GLR test. And they obtained non-parametric version Wilks theorems. In this paper, we extend the GLR test to the index parameter ao in the single-index model, and establish the GLR statistics and demonstrate that its limiting null distribution follows aχ2-distribution. This not only unveils a new type of Wilks phenomenon for the semiparametric regression models with indexes, but also enlarges the application of the GLR test. A simulated example is used to illustrate the good performance of the testing approach.Secondly statistical estimating and testing for the partially linear single-index model is considered in this paper. It mainly includes the profile least-squared estimators of the model and the checking for the index parameter and the index function of the model. The partially linear single-index model proposed by Carroll et al. (1997) is the generalization of the single-index model, and it can be written as where Z= (Z1,…,ZP)T (?) Rp are covariates, the unknown parameter vectorβ0= (β01,…,β0p)T is in RP, the errorεis independent of X and Z, and the other conditions is the same as that used in the above single-index model. A profile least-squares technique for estimating the parametric part is proposed and the asymptotic normality of the profile least-squares estimator is given. Based on the estimator, the GLR test is proposed to test whether parameters on linear part and the index parameter for the model are under a contain linear restricted condition. Under the null model, the proposed GLR statistic follows asymptotically the x2-distribution, which unveils the Wilks phenomenon. The simulated data example shows that our proposed methods is very well.Thirdly tests for the varying-coefficient single-index model are considered in this paper. It mainly includes the tests for the index parameter, the index function and the coefficient functions of the model. To study the relationship between the levels of chemical pollutants and the number of daily total hospital admissions for respiratory diseases, and consider the five pollutants and two environmental factors, i.e. sulphur dioxide, nitrogen dioxide, nitrogen oxide, respirable suspended particulate, ozone, temperature and relative humidity, Wong et al. (2008) proposed the varying-coefficient single-index model where a(·)= (α1(·),…,αp(·))T is an unknown function vector, U (?) R is a covariate,the errorεis independent of X and Z, and the other conditions is the same as that used in the above single-index model. They discussed the relationship between the levels of chemical pollutants and the number of daily total hospital admissions for respiratory diseases. Note that the model has two different covariates in the index function and functional coefficient, which makes it very difficult to estimate and test the model. Wong et al. (2008) employed the local linear method, the average method and the back-fitting technique to obtain the estimates of the unknown parameters and the unknown functions of the model, and gave their asymptotic distribution. They also employed the model to the application of the public health. In this paper, we consider the GLR test on the model, which answers the questions that whether the index function is the linear function and the coefficient functions are not varying. And the inference for the index parameterα0 is also considered. We prove that the proposed test statistics follow theχ2-distribution asymptotically, which unveils a new type of Wilks phenomenon, and both simulated and real examples are used to show the good performance of the testing approach.