Shrinkage estimation of partially linear single-index models

In this paper, we propose shrinkage estimation for partially linear single-index models. A profile least squares approximation is used to estimate the model parameters and select informative variables simultaneously. The resulting estimator is shown to be consistent and to enjoy the oracle properties.     

Semi-parametric estimation of partially linear single-index models

One of the most difficult problems in applications of semi-parametric partially linear single-index models (PLSIM) is the choice of pilot estimators and complexity parameters which may result in radically different estimators. Pilot estimators are often assumed to be root-n consistent, although they are not given in a constructible way. Complexity parameters, such as a smoothing bandwidth are constrained to a certain speed, which is rarely determinable in practical situations.In this paper, efficient, constructible and practicable estimators of PLSIMs are designed with applications to time series. The proposed technique answers two questions from Carroll et al. [Generalized partially linear single-index models, J. Amer. Statist. Assoc. 92 (1997) 477-489]: no root-n pilot estimator for the single-index part of the model is needed and complexity parameters can be selected at the optimal smoothing rate. The asymptotic distribution is derived and the corresponding algorithm is easily implemented. Examples from real data sets (credit-scoring and environmental statistics) illustrate the technique and the proposed methodology of minimum average variance estimation (MAVE).     

Empirical likelihood inference in partially linear single-index models for longitudinal data

The empirical likelihood method is especially useful for constructing confidence intervals or regions of parameters of interest. Yet, the technique cannot be directly applied to partially linear single-index models for longitudinal data due to the within-subject correlation. In this paper, a bias-corrected block empirical likelihood (BCBEL) method is suggested to study the models by accounting for the within-subject correlation. BCBEL shares some desired features: unlike any normal approximation based method for confidence region, the estimation of parameters with the iterative algorithm is avoided and a consistent estimator of the asymptotic covariance matrix is not needed. Because of bias correction, the BCBEL ratio is asymptotically chi-squared, and hence it can be directly used to construct confidence regions of the parameters without any extra Monte Carlo approximation that is needed when bias correction is not applied. The proposed method can naturally be applied to deal with pure single-index models and partially linear models for longitudinal data. Some simulation studies are carried out and an example in epidemiology is given for illustration.     

Randomly censored partially linear single-index models

This paper proposes a method for estimation of a class of partially linear single-index models with randomly censored samples. The method provides a flexible way for modelling the association between a response and a set of predictor variables when the response variable is randomly censored. It presents a technique for “dimension reduction” in semiparametric censored regression models and generalizes the existing accelerated failure-time models for survival analysis. The estimation procedure involves three stages: first, transform the censored data into synthetic data or pseudo-responses unbiasedly; second, obtain quasi-likelihood estimates of the regression coefficients in both linear and single-index components by an iteratively algorithm; finally, estimate the unknown nonparametric regression function using techniques for univariate censored nonparametric regression. The estimators for the regression coefficients are shown to be jointly root-n consistent and asymptotically normal. In addition, the estimator for the unknown regression function is a local linear kernel regression estimator and can be estimated with the same efficiency as all the parameters are known. Monte Carlo simulations are conducted to illustrate the proposed methodology.     

Corrected empirical likelihood inference for right-censored partially linear single-index model

This article deals with the inference on a right-censored partially linear single-index model (RCPLSIM). The main focus is the local empirical likelihood-based inference on the nonparametric part in RCPLSIM. With a synthetic data approach, an empirical log-likelihood ratio statistic for the nonparametric part is defined and it is shown that its limiting distribution is not a central chi-squared distribution. To increase the accuracy of the confidence interval, we also propose a corrected empirical log-likelihood ratio statistic for the nonparametric function. The resulting statistic is proved to follow a standard chi-squared limiting distribution. Simulation studies are undertaken to assess the finite sample performance of the proposed confidence intervals. A real example is also considered.     

Empirical likelihood for single-index models

The empirical likelihood method is especially useful for constructing confidence intervals or regions of the parameter of interest. This method has been extensively applied to linear regression and generalized linear regression models. In this paper, the empirical likelihood method for single-index regression models is studied. An estimated empirical log-likelihood approach to construct the confidence region of the regression parameter is developed. An adjusted empirical log-likelihood ratio is proved to be asymptotically standard chi-square. A simulation study indicates that compared with a normal approximation-based approach, the proposed method described herein works better in terms of coverage probabilities and areas (lengths) of confidence regions (intervals).     

Empirical likelihood for heteroscedastic partially linear models

We make empirical-likelihood-based inference for the parameters in heteroscedastic partially linear models. Unlike the existing empirical likelihood procedures for heteroscedastic partially linear models, the proposed empirical likelihood is constructed using components of a semiparametric efficient score. We show that it retains the double robustness feature of the semiparametric efficient estimator for the parameters and shares the desirable properties of the empirical likelihood for linear models. Compared with the normal approximation method and the existing empirical likelihood methods, the empirical likelihood method based on the semiparametric efficient score is more attractive not only theoretically but empirically. Simulation studies demonstrate that the proposed empirical likelihood provides smaller confidence regions than that based on semiparametric inefficient estimating equations subject to the same coverage probabilities. Hence, the proposed empirical likelihood is preferred to the normal approximation method as well as the empirical likelihood method based on semiparametric inefficient estimating equations, and it should be useful in practice.     

Estimation in generalized linear models for functional data via penalized likelihood

We analyze in a regression setting the link between a scalar response and a functional predictor by means of a Functional Generalized Linear Model. We first give a theoretical framework and then discuss identifiability of the model. The functional coefficient of the model is estimated via penalized likelihood with spline approximation. The L² rate of convergence of this estimator is given under smoothness assumption on the functional coefficient. Heuristic arguments show how these rates may be improved for some particular frameworks.     

Local Walsh-average-based estimation and variable selection for single-index models

We propose a robust estimation procedure based on local Walsh-average regression(LWR) for single-index models. Our novel method provides a root-n consistent estimate of the single-index parameter under some mild regularity conditions; the estimate of the unknown link function converges at the usual rate for the nonparametric estimation of a univariate covariate. We theoretically demonstrate that the new estimators show significant efficiency gain across a wide spectrum of non-normal error distributions and have almost no loss of efficiency for the normal error. Even in the worst case, the asymptotic relative efficiency(ARE) has a lower bound compared with the least squares(LS) estimates; the lower bounds of the AREs are 0.864 and 0.8896 for the single-index parameter and nonparametric function, respectively. Moreover, the ARE of the proposed LWR-based approach versus the ARE of the LS-based method has an expression that is closely related to the ARE of the signed-rank Wilcoxon test as compared with the t-test. In addition, to obtain a sparse estimate of the single-index parameter, we develop a variable selection procedure by combining the estimation method with smoothly clipped absolute deviation penalty; this procedure is shown to possess the oracle property. We also propose a Bayes information criterion(BIC)-type criterion for selecting the tuning parameter and further prove its ability to consistently identify the true model. We conduct some Monte Carlo simulations and a real data analysis to illustrate the finite sample performance of the proposed methods.     

Empirical likelihood confidence regions of the parameters in a partially linear single-index model

In this paper,a partially linear single-index model is investigated,and three empirical log-likelihood ratio statistics for the unknown parameters in the model are sug- gested.It is proved that the proposed statistics are asymptotically standard chi-square un- der some suitable conditions,and hence can be used to construct the confidence regions of the parameters.Our methods can also deal with the confidence region construction for the index in the pure single-index model.A simulation study indicates that,in terms of cov- erage probabilities and average areas of the confidence regions,the proposed methods perform better than the least-squares method.     

An empirical likelihood method in a partially linear single-index model with right censored data

Empirical-likelihood-based inference for the parameters in a partially linear single-index model with randomly censored data is investigated. We introduce an estimated empirical likelihood for the parameters using a synthetic data approach and show that its limiting distribution is a mixture of central chi-squared distribution. To attack this difficulty we propose an adjusted empirical likelihood to achieve the standard χ2-limit. Furthermore, since the index is of norm 1, we use this constraint to reduce the dimension of parameters, which increases the accuracy of the confidence regions. A simulation study is carried out to compare its finite-sample properties with the existing method. An application to a real data set is illustrated.     

Double penalized variable selection procedure for partially linear models with longitudinal data

Based on the double penalized estimation method,a new variable selection procedure is proposed for partially linear models with longitudinal data.The proposed procedure can avoid the effects of the nonparametric estimator on the variable selection for the parameters components.Under some regularity conditions,the rate of convergence and asymptotic normality of the resulting estimators are established.In addition,to improve efficiency for regression coefficients,the estimation of the working covariance matrix is involved in the proposed iterative algorithm.Some simulation studies are carried out to demonstrate that the proposed method performs well.     

Analysis of correlated binary data under partially linear single-index logistic models

Clustered data arise commonly in practice and it is often of interest to estimate the mean response parameters as well as the association parameters. However, most research has been directed to address the mean response parameters with the association parameters relegated to a nuisance role. There is relatively little work concerning both the marginal and association structures, especially in the semiparametric framework. In this paper, our interest centers on the inference of both the marginal and association parameters. We develop a semiparametric method for clustered binary data and establish the theoretical results. The proposed methodology is investigated through various numerical studies.     

响应变量存在缺失时部分线性模型的经验似然推断

考虑响应变量带有缺失的部分线性模型,采用借补的思想,研究了参数部分和非参数部分的经验似然推断,证明了所提出的经验对数似然比统计量依分布收敛到χ2分布,由此构造参数部分和函数部分的置信域和逐点置信区间.对参数部分,模拟比较了经验似然与正态逼近方法;对函数部分,模拟了函数的逐点置信区间.     

Empirical likelihood for the parametric part in partially linear errors-in-function models

Partially linear errors-in-function models were proposed by Liang (2000), but their inferences have not been systematically studied. This article proposes an empirical likelihood method to construct confidence regions of the parametric components. Under mild regularity conditions, the nonparametric version of the Wilk’s theorem is derived. Simulation studies show that the proposed empirical likelihood method provides narrower confidence regions, as well as higher coverage probabilities than those based on the traditional normal approximation method.     

Robust estimation for partially linear models with large-dimensional covariates

We are concerned with robust estimation procedures to estimate the parameters in partially linear models with large-dimensional covariates. To enhance the interpretability, we suggest implementing a nonconcave regularization method in the robust estimation procedure to select important covariates from the linear component. We establish the consistency for both the linear and the nonlinear components when the covariate dimension diverges at the rate of o(n1/2), where n is the sample size. We show that the robust estimate of linear component performs asymptotically as well as its oracle counterpart which assumes the baseline function and the unimportant covariates were known a priori. With a consistent estimator of the linear component, we estimate the nonparametric component by a robust local linear regression. It is proved that the robust estimate of nonlinear component performs asymptotically as well as if the linear component were known in advance.Comprehensive simulation studies are carried out and an application is presented to examine the fnite-sample performance of the proposed procedures.     

Nonconcave penalized inverse regression in single-index models with high dimensional predictors

In this paper we aim to estimate the direction in general single-index models and to select important variables simultaneously when a diverging number of predictors are involved in regressions. Towards this end, we propose the nonconcave penalized inverse regression method. Specifically, the resulting estimation with the SCAD penalty enjoys an oracle property in semi-parametric models even when the dimension, p_n, of predictors goes to infinity. Under regularity conditions we also achieve the asymptotic normality when the dimension of predictor vector goes to infinity at the rate of p_n=o(n^1/3) where n is sample size, which enables us to construct confidence interval/region for the estimated index. The asymptotic results are augmented by simulations, and illustrated by analysis of an air pollution dataset.     

Empirical likelihood for single-index models with responses missing at random

The missing response problem in single-index models is studied, and a bias-correction method to infer the index coefficients is developed. Two weighted empirical log-likelihood ratios with asymptotic chisquare are derived, and the corresponding empirical likelihood confidence regions for the index coefficients are constructed. In addition, the estimators of the index coefficients and the link function are defined, and their asymptotic normalities are proved. A simulation study is conducted to compare the empirical likelihood and the normal approximation based method in terms of coverage probabilities and average lengths of confidence intervals. A real example illustrates our methods.     

Statistical inference on parametric part for partially linear single-index model

Statistical inference on parametric part for the partially linear single-index model (PLSIM) is considered in this paper. 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, a generalized likelihood ratio (GLR) test is proposed to test whether parameters on linear part for the model is under a contain linear restricted condition. Under the null model, the proposed GLR statistic follows asymptotically the χ²-distribution with the scale constant and degree of freedom independent of the nuisance parameters, known as Wilks phenomenon. Both simulated and real data examples are used to illustrate our proposed methods.