Statistical inferences for partially linear single-index models with error-prone linear covariates

We consider statistical inference for partially linear single-index models (PLSIM) when some linear covariates are not observed, but ancillary variables are available. Based on the profile least-squared estimators of the unknowns, we study the testing problems for parametric components in the proposed models. It is to see whether the generalized likelihood ratio (GLR) tests proposed by Fan et al. (2001) are applicable to testing for the parametric components. We show that under the null hypothesis the proposed GLR statistics follow asymptotically the χ²-distributions with the scale constants and the degrees of freedom being independent of the nuisance parameters or functions, which is called the Wilks phenomenon. Simulated experiments are conducted to illustrate our proposed methodology.     

Empirical-likelihood-based Test for Partially Linear Single-index Models with Error-prone Linear Covariates

In this article, we consider whether the empirical likelihood ratio (ELR) test is applicable to testing for serial correlation in the partially linear single-index models (PLSIM) with error-prone linear covariates. It is shown that under the null hypothesis the proposed ELR statistic follows asymptotically a χ²-distribution with the scale constant and the degrees of freedom. A comparison between the ELR and the normal approximation method is also considered. Both simulated and real data examples are used to illustrate our proposed methodology.     

Empirical likelihood inference in partially linear single-index models with endogenous covariates

In this article, we consider how to construct the confidence regions of the unknown parameters for partially linear single-index models with endogenous covariates. To eliminate the influence of the endogenous covariates, an empirical likelihood method is proposed based on instrumental variables. Under some regularly conditions, the asymptotic distribution of the proposed empirical log-likelihood ratio is proved to be a Chi-squared distribution. We investigate the finite-sample performance of the proposed method via simulation studies.     

Statistical inference on asymptotic properties of two estimators for the partially linear single-index models

The outer product of gradients (OPG) estimation procedure based on least squares (LS) approach has been presented by Xia et al. [An adaptive estimation of dimension reduction space. J Roy Statist Soc Ser B. 2002;64:363–410] to estimate the single-index parameter in partially linear single-index models (PLSIM). However, its asymptotic property has not been established yet and the efficiency of LS-based method can be significantly affected by outliers and heavy-tailed distributions. In this paper, we firstly derive the asymptotic property of OPG estimator developed by Xia et al. [An adaptive estimation of dimension reduction space. J Roy Statist Soc Ser B. 2002;64:363–410] in theory, and a novel robust estimation procedure combining the ideas of OPG and local rank (LR) inference is further developed for PLSIM along with its theoretical property. Then, we theoretically derive the asymptotic relative efficiency (ARE) of the proposed LR-based procedure with respect to LS-based method, which is shown to possess an expression that is closely related to that of the signed-rank Wilcoxon test in comparison with the t-test. Moreover, we demonstrate that the new proposed estimator has a great efficiency gain across a wide spectrum of non-normal error distributions and almost not lose any efficiency for the normal error. Even in the worst case scenarios, the ARE owns a lower bound equalling to 0.864 for estimating the single-index parameter and a lower bound being 0.8896 for estimating the nonparametric function respectively, versus the LS-based estimators. Finally, some Monte Carlo simulations and a real data analysis are conducted to illustrate the finite sample performance of the estimators.     

Robust estimation for partially linear single-index model

In this article, we investigate a new estimation approach for the partially linear single-index model based on modal regression method, where the non parametric function is estimated by penalized spline method. Moreover, we develop an expection maximum (EM)-type algorithm and establish the large sample properties of the proposed estimation method. A distinguishing characteristic of the newly proposed estimation is robust against outliers through introducing an additional tuning parameter which can be automatically selected using the observed data. Simulation studies and real data example are used to evaluate the finite-sample performance, and the results show that the newly proposed method works very well.     

Statistical estimation for a partially linear single-index model with errors in all variables

This article considers partially linear single-index models with errors in all variables. By using the Pseudo ? θ method (Liang, Härdle, and Carroll 1999), local linear regression and simulation-extrapolation (SIMEX) technique (Cook and Stefanski 1994), we propose an efficient methodology to estimate the current model. Under certain conditions the asymptotic properties of proposed estimators are obtained. Some simulation experiments and an application are conducted to illustrate our proposed method.     

Results concerning the generalized partially linear single-index model

The central topic of this article is the estimation of parameters of the generalized partially linear single-index model (GPLSIM). Two numerical optimization procedures are presented and an S-plus program based on these procedures is compared to a program by Wand in a simulation setting. The results from these simulations indicate that the estimates for the new procedures are as good, if not better, than Wand's. Also, this program is much more flexible than Wand's since it can handle more general models. Other simulations are also conducted. The first compares the effects of using linear interpolation versus spline interpolation in an optimization procedure. The results indicate that by using spline interpolation one gets more stable estimates at a cost of increased computational time. A second simulation was conducted to assess the performance of a method for estimating the variance of alpha. A third set of simulations is carried out to determine the best criterion for testing that one of the elements of alpha is equal to zero. The GPLSIM is applied to a water quality data set and the results indicate an interesting relationship between gastrointestinal illness and turbidity (cloudiness) of drinking water.     

Empirical likelihood-based serial correlation testing in partially varying coefficient single-index models

ABSTRACT

Partially varying coefficient single-index models (PVCSIM) are a class of semiparametric regression models. One important assumption is that the model error is independently and identically distributed, which may contradict with the reality in many applications. For example, in the economical and financial applications, the observations may be serially correlated over time. Based on the empirical likelihood technique, we propose a procedure for testing the serial correlation of random error in PVCSIM. Under some regular conditions, we show that the proposed empirical likelihood ratio statistic asymptotically follows a standard χ² distribution. We also present some numerical studies to illustrate the performance of our proposed testing procedure.     

Semiparametric inference with a functional-form empirical likelihood

A functional-form empirical likelihood method is proposed as an alternative method to the empirical likelihood method. The proposed method has the same asymptotic properties as the empirical likelihood method but has more flexibility in choosing the weight construction. Because it enjoys the likelihood-based interpretation, the profile likelihood ratio test can easily be constructed with a chi-square limiting distribution. Some computational details are also discussed, and results from finite-sample simulation studies are presented.     

Comparison of Nonparametric Components in Two Partially Linear Models

In this article, we are concerned with whether the nonparametric functions are parallel from two partial linear models, and propose a test statistic to check the difference of the two functions. The unknown constant α is estimated by using moment method under null models. Nonparametric functions under both null and full models are estimated by using local linear method. The asymptotic properties of parametric and nonparametric components are derived. The test statistic under the null hypothesis is calculated and shown to be asymptotically normal.     

Generalized empirical likelihood inference in partially linear model for longitudinal data with missing response variables and error-prone covariates

In this article, we consider statistical inference for longitudinal partial linear models when the response variable is sometimes missing with missingness probability depending on the covariate that is measured with error. A generalized empirical likelihood (GEL) method is proposed by combining correction attenuation and quadratic inference functions. The method that takes into consideration the correlation within groups is used to estimate the regression coefficients. Furthermore, residual-adjusted empirical likelihood (EL) is employed for estimating the baseline function so that undersmoothing is avoided. The empirical log-likelihood ratios are proven to be asymptotically Chi-squared, and the corresponding confidence regions for the parameters of interest are then constructed. Compared with methods based on NAs, the GEL does not require consistent estimators for the asymptotic variance and bias. The numerical study is conducted to compare the performance of the EL and the normal approximation-based method, and a real example is analysed.     

Estimation of partially linear single-index additive hazards model with current status data

In this paper, we introduce a partially linear single-index additive hazards model with current status data. Both the unknown link function of the single-index term and the cumulative baseline hazard function are approximated by B-splines under a monotonicity constraint on the latter. The sieve method is applied to estimate the nonparametric and parametric components simultaneously. We show that, when the nonparametric link function is an exact B-spline, the resultant estimator of regression parameter vector is asymptotically normal and achieves the semiparametric information bound and the rate of convergence of the estimator for the cumulative baseline hazard function is optimal. Simulation studies are presented to examine the finite sample performance of the proposed estimation method. For illustration, we apply the method to a clinical dataset with current status outcome.     

Estimation and inference for varying coefficient partially nonlinear models

In this paper, we extend the varying coefficient partially linear model to the varying coefficient partially nonlinear model in which the linear part of the varying coefficient partially linear model is replaced by a nonlinear function of the covariates. A profile nonlinear least squares estimation procedure for the parameter vector and the coefficient function vector of the varying coefficient partially nonlinear model is proposed and the asymptotic properties of the resulting estimators are established. We further propose a generalized likelihood ratio (GLR) test to check whether or not the varying coefficients in the model are constant. The asymptotic null distribution of the GLR statistic is derived and a residual-based bootstrap procedure is also suggested to derive the p-value of the GLR test. Some simulations are conducted to assess the performance of the proposed estimating and testing procedures and the results show that both the procedures perform well in finite samples. Furthermore, a real data example is given to demonstrate the application of the proposed model and its estimating and testing procedures.     

Empirical likelihood based diagnostics for heteroscedasticity in partially linear errors-in-variables models

A standard assumption in regression analysis is homogeneity of the error variance. Violation of this assumption can have adverse consequences for the efficiency of estimators. In this paper, we propose an empirical likelihood based diagnostic technique for heteroscedasticity in the partially linear errors-in-variables models. Under mild conditions, a nonparametric version of Wilk's theorem is derived. Simulation results reveal that our test performs well in both size and power.     

Statistical inference in partially time-varying coefficient models

A partially time-varying coefficient time series model is introduced to characterize the nonlinearity and trending phenomenon. To estimate the regression parameter and the nonlinear coefficient function, the profile least squares approach is applied with the help of local linear approximation. The asymptotic distributions of the proposed estimators are established under mild conditions. Meanwhile, the generalized likelihood ratio test is studied and the test statistics are demonstrated to follow asymptotic χ²-distribution under the null hypothesis. Furthermore, some extensions of the proposed model are discussed and several numerical examples are provided to illustrate the finite sample behavior of the proposed methods.     

Estimation and inference for varying coefficient partially nonlinear errors-in-variables models

In this article, we study the varying coefficient partially nonlinear model with measurement errors in the nonparametric part. A local corrected profile nonlinear least-square estimation procedure is proposed and the asymptotic properties of the resulting estimators are established. Further, a generalized likelihood ratio (GLR) statistic is proposed to test whether the varying coefficients are constant. The asymptotic null distribution of the statistic is obtained and a residual-based bootstrap procedure is employed to compute the p-value of the statistic. Some simulations are conducted to evaluate the performance of the proposed methods. The results show that the estimating and testing procedures work well in finite samples.     

Sensitivity analysis of partially linear models with response missing at random

This article investigates case-deletion influence analysis via Cook’s distance and local influence analysis via conformal normal curvature for partially linear models with response missing at random. Local influence approach is developed to assess the sensitivity of parameter and nonparametric estimators to various perturbations such as case-weight, response variable, explanatory variable, and parameter perturbations on the basis of semiparametric estimating equations, which are constructed using the inverse probability weighted approach, rather than likelihood function. Residual and generalized leverage are also defined. Simulation studies and a dataset taken from the AIDS Clinical Trials are used to illustrate the proposed methods.     

Testing Serial Correlation in Partially Linear Single-Index Errors-in-Variables Models

In this article, we propose two test statistics for testing the underlying serial correlation in a partially linear single-index model Y = η(Z ^τα) + X ^τβ + ? when X is measured with additive error. The proposed test statistics are shown to have asymptotic normal or chi-squared distributions under the null hypothesis of no serial correlation. Monte Carlo experiments are also conducted to illustrate the finite sample performance of the proposed test statistics. The simulation results confirm that these statistics perform satisfactorily in both estimated sizes and powers.     

Inference for Nonparametric Parts in Single-Index Varying-Coefficient Model

Xia and Li (1999 Xia , Y. , Li , W. ( 1999 ). On single-index coefficient regression models . J. Amer. Statist. Assoc. 94 : 1275 – 1284 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) proposed the single-index varying-coefficient model (SIVCM), which is frequently used in statistical modeling. However, the inference for the SIVCM has not been very well developed. In this article, our main purpose is to examine whether the generalized likelihood ratio (GLR) test is applicable to the testing problem for the nonparametric parts of the SIVCMs. Under the null hypothesis the newly proposed GLR statistic asymptotically follows the chi-squared distribution with scale constant and degree of freedom independent of the nuisance parameters or functions. A new Wilks phenomenon is unveiled. A simulated example is given to evaluate our proposed methods.     

The Empirical Likelihood Goodness-of-Fit Test for a Regression Model with Randomly Censored Data

The regression model with randomly censored data has been intensively investigated. In this article, we consider a goodness-of-fit test for this model. Empirical likelihood (EL) tests are constructed. The asymptotic distributions of the test statistic under null hypothesis and the local alternative hypothesis are given. Simulations are carried out to illustrate the methodology.