Riquan Zhang

Varying-coefficient single-index model

In this paper, the varying-coefficient single-index model (VCSIM) is proposed. It can be seen as a generalization of the semivarying-coefficient 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.

A robust and efficient estimation method for single index models

Single index models are natural extensions of linear models and overcome the so-called curse of dimensionality. They have applications to many fields, such as medicine, economics and finance. However, most existing methods based on least squares or likelihood are sensitive when there are outliers or the error distribution is heavy tailed. Although an M-type regression is often considered as a good alternative to those methods, it may lose efficiency for normal errors. In this paper, we propose a new robust and efficient estimation procedure based on local modal regression for single index models. The asymptotic normality of proposed estimators for both the parametric and nonparametric parts is established. We show that the proposed estimators are as asymptotically efficient as the least-square-based estimators when there are no outliers and the error distribution is normal. A modified EM algorithm is presented for efficient implementation. The simulations and real data analysis are conducted to illustrate the finite sample performance of the proposed method.

Profile empirical-likelihood inferences for the single-index-coefficient regression model

This article deals with a new profile empirical-likelihood inference for a class of frequently used single-index-coefficient regression models (SICRM), which were proposed by Xia and Li (J. Am. Stat. Assoc. 94:1275–1285, 1999a). Applying the empirical likelihood method (Owen in Biometrika 75:237–249, 1988), a new estimated empirical log-likelihood ratio statistic for the index parameter of the SICRM is proposed. To increase the accuracy of the confidence region, a new profile empirical likelihood for each component of the relevant parameter is obtained by using maximum empirical likelihood estimators (MELE) based on a new and simple estimating equation for the parameters in the SICRM. Hence, the empirical likelihood confidence interval for each component is investigated. Furthermore, corrected empirical likelihoods for functional components are also considered. The resulting statistics are shown to be asymptotically standard chi-squared distributed. Simulation studies are undertaken to assess the finite sample performance of our method. A study of real data is also reported.

Generalized likelihood ratio test for varying-coefficient models with different smoothing variables

Varying-coefficient models are popular multivariate nonparametric fitting techniques. When all coefficient functions in a varying-coefficient 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 @g^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.

Variable selection for varying dispersion beta regression model

The beta regression models are commonly used by practitioners to model variables that assume values in the standard unit interval (0, 1). In this paper, we consider the issue of variable selection for beta regression models with varying dispersion (VBRM), in which both the mean and the dispersion depend upon predictor variables. Based on a penalized likelihood method, the consistency and the oracle property of the penalized estimators are established. Following the coordinate descent algorithm idea of generalized linear models, we develop new variable selection procedure for the VBRM, which can efficiently simultaneously estimate and select important variables in both mean model and dispersion model. Simulation studies and body fat data analysis are presented to illustrate the proposed methods.

Statistical estimation in varying coefficient models with surrogate data and validation sampling

Varying coefficient error-in-covariables models are considered with surrogate data and validation sampling. Without specifying any error structure equation, two estimators for the coefficient function vector are suggested by using the local linear kernel smoothing technique. The proposed estimators are proved to be asymptotically normal. A bootstrap procedure is suggested to estimate the asymptotic variances. The data-driven bandwidth selection method is discussed. A simulation study is conducted to evaluate the proposed estimating methods.

Quantile regression and variable selection for the single-index model

In this paper, we propose a new full iteration estimation method for quantile regression (QR) of the single-index model (SIM). The asymptotic properties of the proposed estimator are derived. Furthermore, we propose a variable selection procedure for the QR of SIM by combining the estimation method with the adaptive LASSO penalized method to get sparse estimation of the index parameter. The oracle properties of the variable selection method are established. Simulations with various non-normal errors are conducted to demonstrate the finite sample performance of the estimation method and the variable selection procedure. Furthermore, we illustrate the proposed method by analyzing a real data set.

TESTS FOR VARYING-COEFFICIENT PARTS ON VARYING-COEFFICIENT SINGLE-INDEX MODEL

To study the relationship between the levels of chemical pol- lutants and the number of daily total hospital admissions for respiratory diseases and to find the eect of temperature/relative humidity on the admission number, Wong et al. (17) introduced the varying-coecient single-index model (VCSIM). As pointed out, it is a popular multivari- ate nonparametric fitting technique. However, the tests of the model have not been very well developed. In this paper, based on the estima- tors 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-coecient parts on the VCSIM are estab- lished. Under the null hypotheses the new proposed GLR tests follow the ´ 2 -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.

Partially varying coefficient single index proportional hazards regression models

In this paper, the partially varying coefficient single index proportional hazards regression models are discussed. All unknown functions are fitted by polynomial B splines. The index parameters and B-spline coefficients are estimated by the partial likelihood method and a two-step Newton-Raphson algorithm. Consistency and asymptotic normality of the estimators of all the parameters are derived. Through a simulation study and the VA data example, we illustrate that the proposed estimation procedure is accurate, rapid and stable.

Efficient empirical-likelihood-based inferences for the single-index model

This article proposes the efficient empirical-likelihood-based inferences for the single component of the parameter and the link function in the single-index model. Unlike the existing empirical likelihood procedures for the single-index model, the proposed profile empirical likelihood for the parameter is constructed by using some components of the maximum empirical likelihood estimator (MELE) based on a semiparametric efficient score. The empirical-likelihood-based inference for the link function is also considered. The resulting statistics are proved to follow a standard chi-squared limiting distribution. Simulation studies are undertaken to assess the finite sample performance of the proposed confidence intervals. An application to real data set is illustrated.