Functional-coefficient partially linear regression model

In this paper, the functional-coefficient partially linear regression (FCPLR) model is proposed by combining nonparametric and functional-coefficient regression (FCR) model. It includes the FCR model and the nonparametric regression (NPR) model as its special cases. It is also a generalization of the partially linear regression (PLR) model obtained by replacing the parameters in the PLR model with some functions of the covariates. The local linear technique and the integrated method are employed to give initial estimators of all functions in the FCPLR model. These initial estimators are asymptotically normal. The initial estimator of the constant part function shares the same bias as the local linear estimator of this function in the univariate nonparametric model, but the variance of the former is bigger than that of the latter. Similarly, initial estimators of every coefficient function share the same bias as the local linear estimates in the univariate FCR model, but the variance of the former is bigger than that of the latter. To decrease the variance of the initial estimates, a one-step back-fitting technique is used to obtain the improved estimators of all functions. The improved estimator of the constant part function has the same asymptotic normality property as the local linear nonparametric regression for univariate data. The improved estimators of the coefficient functions have the same asymptotic normality properties as the local linear estimates in FCR model. The bandwidths and the smoothing variables are selected by a data-driven method. Both simulated and real data examples related to nonlinear time series modeling are used to illustrate the applications of the FCPLR model.     

Statistical inference in partially-varying-coefficient single-index model

Consider a varying-coefficient single-index model which consists of two parts: the linear part with varying coefficients and the nonlinear part with a single-index structure, and are hence termed as varying-coefficient single-index models. This model includes many important regression models such as single-index models, partially linear single-index models, varying-coefficient model and varying-coefficient partially linear models as special examples. In this paper, we mainly study estimating problems of the varying-coefficient vector, the nonparametric link function and the unknown parametric vector describing the single-index in the model. A stepwise approach is developed to obtain asymptotic normality estimators of the varying-coefficient vector and the parametric vector, and estimators of the nonparametric link function with a convergence rate. The consistent estimator of the structural error variance is also obtained. In addition, asymptotic pointwise confidence intervals and confidence regions are constructed for the varying coefficients and the parametric vector. The bandwidth selection problem is also considered. A simulation study is conducted to evaluate the proposed methods, and real data analysis is also used to illustrate our methods.     

Asymptotic distributions of two “synthetic data” estimators for censored single-index models

The censored single-index model provides a flexible way for modelling the association between a response and a set of predictor variables when the response variable is randomly censored and the link function is unknown. It presents a technique for “dimension reduction” in semiparametric censored regression models and generalizes the existing accelerated failure time models for survival analysis. This paper proposes two methods for estimation of single-index models with randomly censored samples. We first transform the censored data into synthetic data or pseudo-responses unbiasedly, then obtain estimates of the index coefficients by the rOPG or rMAVE procedures of Xia (2006) [1]. Finally, we estimate the unknown nonparametric link function using techniques for univariate censored nonparametric regression. The estimators for the index coefficients are shown to be 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 the parameters are known. Monte Carlo simulations are conducted to illustrate the proposed methodologies.     

Nonparametric variance function estimation with missing data

In this paper, a fixed design regression model where the errors follow a strictly stationary process is considered. In this model the conditional mean function and the conditional variance function are unknown curves. Correlated errors when observations are missing in the response variable are assumed. Four nonparametric estimators of the conditional variance function based on local polynomial fitting are proposed. Expressions of the asymptotic bias and variance of these estimators are obtained. A simulation study illustrates the behavior of the proposed estimators.     

Probability density estimation for survival data with censoring indicators missing at random

In this paper, some nonparametric approaches of density function estimation are developed when censoring indicators are missing at random. A conditional mean score based estimator and a mean score estimator are suggested, respectively. The two estimators are proved to be asymptotically normal and uniformly strongly consistent. The bandwidth selection problem is also discussed. A simulation study is conducted to compare finite-sample behaviors of the proposed estimators.     

Censored multiple regression by the method of average derivatives

This paper proposes a technique [termed censored average derivative estimation (CADE)] for studying estimation of the unknown regression function in nonparametric censored regression models with randomly censored samples. The CADE procedure involves three stages: firstly-transform the censored data into synthetic data or pseudo-responses using the inverse probability censoring weighted (IPCW) technique, secondly estimate the average derivatives of the regression function, and finally approximate the unknown regression function by an estimator of univariate regression using techniques for one-dimensional nonparametric censored regression. The CADE provides an easily implemented methodology for modelling the association between the response and a set of predictor variables when data are randomly censored. It also provides a technique for “dimension reduction” in nonparametric censored regression models. The average derivative estimator is shown to be root-n consistent and asymptotically normal. The estimator of the unknown regression function is a local linear kernel regression estimator and is shown to converge at the optimal one-dimensional nonparametric rate. Monte Carlo experiments show that the proposed estimators work quite well.     

Generalized partially linear models with missing covariates

In this article we study a semiparametric generalized partially linear model when the covariates are missing at random. We propose combining local linear regression with the local quasilikelihood technique and weighted estimating equation to estimate the parameters and nonparameters when the missing probability is known or unknown. We establish normality of the estimators of the parameter and asymptotic expansion for the estimators of the nonparametric part. We apply the proposed models and methods to a study of the relation between virologic and immunologic responses in AIDS clinical trials, in which virologic response is classified into binary variables. We also give simulation results to illustrate our approach.     

Model checking in errors-in-variables regression

This paper discusses a class of minimum distance tests for fitting a parametric regression model to a class of regression functions in the errors-in-variables model. These tests are based on certain minimized distances between a nonparametric regression function estimator and a deconvolution kernel estimator of the conditional expectation of the parametric model being fitted. The paper establishes the asymptotic normality of the proposed test statistics under the null hypothesis and that of the corresponding minimum distance estimators. We also prove the consistency of the proposed tests against a fixed alternative and obtain the asymptotic distributions for general local alternatives. Simulation studies show that the testing procedures are quite satisfactory in the preservation of the finite sample level and in terms of a power comparison.     

Statistical inference in a panel data semiparametric regression model with serially correlated errors

We consider a panel data semiparametric partially linear regression model with an unknown vector β of regression coefficients, an unknown nonparametric function g(·) for nonlinear component, and unobservable serially correlated errors. The correlated errors are modeled by a vector autoregressive process which involves a constant intraclass correlation. Applying the pilot estimators of β and g(·), we construct estimators of the autoregressive coefficients, the intraclass correlation and the error variance, and investigate their asymptotic properties. Fitting the error structure results in a new semiparametric two-step estimator of β, which is shown to be asymptotically more efficient than the usual semiparametric least squares estimator in terms of asymptotic covariance matrix. Asymptotic normality of this new estimator is established, and a consistent estimator of its asymptotic covariance matrix is presented. Furthermore, a corresponding estimator of g(·) is also provided. These results can be used to make asymptotically efficient statistical inference. Some simulation studies are conducted to illustrate the finite sample performances of these proposed estimators.     

A profile-type smoothed score function for a varying coefficient partially linear model

The varying coefficient partially linear model is considered in this paper. When the plug-in estimators of coefficient functions are used, the resulting smoothing score function becomes biased due to the slow convergence rate of nonparametric estimations. To reduce the bias of the resulting smoothing score function, a profile-type smoothed score function is proposed to draw inferences on the parameters of interest without using the quasi-likelihood framework, the least favorable curve, a higher order kernel or under-smoothing. The resulting profile-type statistic is still asymptotically Chi-squared under some regularity conditions. The results are then used to construct confidence regions for the parameters of interest. A simulation study is carried out to assess the performance of the proposed method and to compare it with the profile least-squares method. A real dataset is analyzed for illustration.     

Nonparametric regression function estimation with surrogate data and validation sampling

This paper develops estimation approaches for nonparametric regression analysis with surrogate data and validation sampling when response variables are measured with errors. Without assuming any error model structure between the true responses and the surrogate variables, a regression calibration kernel regression estimate is defined with the help of validation data. The proposed estimator is proved to be asymptotically normal and the convergence rate is also derived. A simulation study is conducted to compare the proposed estimators with the standard Nadaraya-Watson estimators with the true observations in the validation data set and the complete observations, respectively. The Nadaraya-Watson estimator with the complete observations can serve as a gold standard, even though it is practically unachievable because of the measurement errors.     

Nonparametric additive model-assisted estimation for survey data

An additive model-assisted nonparametric method is investigated to estimate the finite population totals of massive survey data with the aid of auxiliary information. A class of estimators is proposed to improve the precision of the well known Horvitz-Thompson estimators by combining the spline and local polynomial smoothing methods. These estimators are calibrated, asymptotically design-unbiased, consistent, normal and robust in the sense of asymptotically attaining the Godambe-Joshi lower bound to the anticipated variance. A consistent model selection procedure is further developed to select the significant auxiliary variables. The proposed method is sufficiently fast to analyze large survey data of high dimension within seconds. The performance of the proposed method is assessed empirically via simulation studies.     

Empirical likelihood for semiparametric regression model with missing response data

A bias-corrected technique for constructing the empirical likelihood ratio is used to study a semiparametric regression model with missing response data. We are interested in inference for the regression coefficients, the baseline function and the response mean. A class of empirical likelihood ratio functions for the parameters of interest is defined so that undersmoothing for estimating the baseline function is avoided. The existing data-driven algorithm is also valid for selecting an optimal bandwidth. Our approach is to directly calibrate the empirical log-likelihood ratio so that the resulting ratio is asymptotically chi-squared. Also, a class of estimators for the parameters of interest is constructed, their asymptotic distributions are obtained, and consistent estimators of asymptotic bias and variance are provided. Our results can be used to construct confidence intervals and bands for the parameters of interest. A simulation study is undertaken to compare the empirical likelihood with the normal approximation-based method in terms of coverage accuracies and average lengths of confidence intervals. An example for an AIDS clinical trial data set is used for illustrating our methods.     

Covariate selection for semiparametric hazard function regression models

We study a flexible class of nonproportional hazard function regression models in which the influence of the covariates splits into the sum of a parametric part and a time-dependent nonparametric part. We develop a method of covariate selection for the parametric part by adjusting for the implicit fitting of the nonparametric part. Asymptotic consistency of the proposed covariate selection method is established, leading to asymptotically normal estimators of both parametric and nonparametric parts of the model in the presence of covariate selection. The approach is applied to a real data set and a simulation study is presented.     

Properties of nonparametric estimators of autocovariance for stationary random fields

Summary We introduce nonparametric estimators of the autocovariance of a stationary random field. One of our estimators has the property that it is itself an autocovatiance. This feature enables the estimator to be used as the basis of simulation studies such as those which are necessary when constructing bootstrap confidence intervals for unknown parameters. Unlike estimators proposed recently by other authors, our own do not require assumptions such as isotropy or monotonicity. Indeed, like nonparametric function estimators considered more widely in the context of curve estimation, our approach demands only smoothness and tail conditions on the underlying curve or surface (here, the autocovariance), and moment and mixing conditions on the random field. We show that by imposing the condition that the estimator be a covariance function we actually reduce the numerical value of integrated squared error.     

Strong consistency of nearest neighbor regression function estimators

For a well-known class of nonparametric regression function estimators of nearest neighbor type the uniform measure of deviation from the estimators to the true regression function is studied. Under weak regularity conditions it is shown that the estimators are uniformly consistent with probability one and the corresponding rate of convergence is near-optimal.     

Variable selection for semiparametric varying coefficient partially linear errors-in-variables models

This paper focuses on the variable selections for semiparametric varying coefficient partially linear models when the covariates in the parametric and nonparametric components are all measured with errors. A bias-corrected variable selection procedure is proposed by combining basis function approximations with shrinkage estimations. With appropriate selection of the tuning parameters, the consistency of the variable selection procedure and the oracle property of the regularized estimators are established. A simulation study and a real data application are undertaken to evaluate the finite sample performance of the proposed method.     

Rates of convergence for minimum contrast estimators

Summary We shall present here a general study of minimum contrast estimators in a nonparametric setting (although our results are also valid in the classical parametric case) for independent observations. These estimators include many of the most popular estimators in various situations such as maximum likelihood estimators, least squares and other estimators of the regression function, estimators for mixture models or deconvolution... The main theorem relates the rate of convergence of those estimators to the entropy structure of the space of parameters. Optimal rates depending on entropy conditions are already known, at least for some of the models involved, and they agree with what we get for minimum contrast estimators as long as the entropy counts are not too large. But, under some circumstances (large entropies or changes in the entropy structure due to local perturbations), the resulting the rates are only suboptimal. Counterexamples are constructed which show that the phenomenon is real for non-parametric maximum likelihood or regression. This proves that, under purely metric assumptions, our theorem is optimal and that minimum contrast estimators happen to be suboptimal.