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.     

Empirical likelihood for a varying coefficient partially linear model with diverging number of parameters

The purpose of this paper is two-fold. First, for the estimation or inference about the parameters of interest in semiparametric models, the commonly used plug-in estimation for infinite-dimensional nuisance parameter creates non-negligible bias, and the least favorable curve or under-smoothing is popularly employed for bias reduction in the literature. To avoid such strong structure assumptions on the models and inconvenience of estimation implementation, for the diverging number of parameters in a varying coefficient partially linear model, we adopt a bias-corrected empirical likelihood (BCEL) in this paper. This method results in the distribution of the empirical likelihood ratio to be asymptotically tractable. It can then be directly applied to construct confidence region for the parameters of interest. Second, different from all existing methods that impose strong conditions to ensure consistency of estimation when diverging the number of the parameters goes to infinity as the sample size goes to infinity, we provide techniques to show that, other than the usual regularity conditions, the consistency holds under moment conditions alone on the covariates and error with a diverging rate being even faster than those in the literature. 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.     

Non-parametric kernel regression for multinomial data

This paper presents a kernel smoothing method for multinomial regression. A class of estimators of the regression functions is constructed by minimizing a localized power-divergence measure. These estimators include the bandwidth and a single parameter originating in the power-divergence measure as smoothing parameters. An asymptotic theory for the estimators is developed and the bias-adjusted estimators are obtained. A data-based algorithm for selecting the smoothing parameters is also proposed. Simulation results reveal that the proposed algorithm works efficiently.     

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.     

Bias-corrected empirical likelihood in a multi-link semiparametric model

In this paper, we investigate the empirical likelihood for constructing a confidence region of the parameter of interest in a multi-link semiparametric model when an infinite-dimensional nuisance parameter exists. The new model covers the commonly used varying coefficient, generalized linear, single-index, multi-index, hazard regression models and their generalizations, as its special cases. Because of the existence of the infinite-dimensional nuisance parameter, the classical empirical likelihood with plug-in estimation cannot be asymptotically distribution-free, and the existing bias correction is not extendable to handle such a general model. We then propose a link-based correction approach to solve this problem. This approach gives a general rule of bias correction via an inner link, and consists of two parts. For the model whose estimating equation contains the score functions that are easy to estimate, we use a centering for the scores to correct the bias; for the model of which the score functions are of complex structure, a bias-correction procedure using simpler functions instead of the scores is given without loss of asymptotic efficiency. The resulting empirical likelihood shares the desired features: it has a chi-square limit and, under-smoothing technique, high order kernel and parameter estimation are not needed. Simulation studies are carried out to examine the performance of the new method.     

Semiparametric analysis of longitudinal zero-inflated count data

In this article, we consider a semiparametric zero-inflated Poisson mixed model that postulates a possible nonlinear relationship between the natural logarithm of the mean of the counts and a particular covariate in the longitudinal studies. A penalized log-likelihood function is proposed and Monte Carlo expectation-maximization algorithm is used to derive the estimates. Under some mild conditions, we establish the consistency and asymptotic normality of the resulting estimators. Simulation studies are carried out to investigate the finite sample performance of the proposed method. For illustration purposes, the method is applied to a data set from a pharmaceutical company where the variable of interest is the number of episodes of side effects after the patient has taken the treatments.     

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.     

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.     

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.     

Some theoretical properties of Silverman's method for Smoothed functional principal component analysis

Principal component analysis (PCA) is one of the key techniques in functional data analysis. One important feature of functional PCA is that there is a need for smoothing or regularizing of the estimated principal component curves. Silverman’s method for smoothed functional principal component analysis is an important approach in a situation where the sample curves are fully observed due to its theoretical and practical advantages. However, lack of knowledge about the theoretical properties of this method makes it difficult to generalize it to the situation where the sample curves are only observed at discrete time points. In this paper, we first establish the existence of the solutions of the successive optimization problems in this method. We then provide upper bounds for the bias parts of the estimation errors for both eigenvalues and eigenfunctions. We also prove functional central limit theorems for the variation parts of the estimation errors. As a corollary, we give the convergence rates of the estimations for eigenvalues and eigenfunctions, where these rates depend on both the sample size and the smoothing parameters. Under some conditions on the convergence rates of the smoothing parameters, we can prove the asymptotic normalities of the estimations.     

A Berry-Esseen type bound in kernel density estimation for strong mixing censored samples

In this paper, we discuss the estimation of a density function based on censored data by the kernel smoothing method when the survival and the censoring times form a stationary α-mixing sequence. A Berry-Esseen type bound is derived for the kernel density estimator at a fixed point x. For practical purposes, a randomly weighted estimator of the density function is also constructed and investigated.     

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.     

Semiparametric analysis in double-sampling designs via empirical likelihood

Double-sampling designs are commonly used in real applications when it is infeasible to collect exact measurements on all variables of interest. Two samples, a primary sample on proxy measures and a validation subsample on exact measures, are available in these designs. We assume that the validation sample is drawn from the primary sample by the Bernoulli sampling with equal selection probability. An empirical likelihood based approach is proposed to estimate the parameters of interest. By allowing the number of constraints to grow as the sample size goes to infinity, the resulting maximum empirical likelihood estimator is asymptotically normal and its limiting variance-covariance matrix reaches the semiparametric efficiency bound. Moreover, the Wilks-type result of convergence to chi-squared distribution for the empirical likelihood ratio based test is established. Some simulation studies are carried out to assess the finite sample performances of the new approach.     

Multivariate Failure Times Regression with a Continuous Auxiliary Covariate

How to take advantage of the available auxiliary covariate information when the primary covariate of interest is not measured is a frequently encountered question in biomedical study. In this paper, we consider the multivariate failure times regression analysis in which the primary covariate is assessed only in a validation set, but a continuous auxiliary covariate for it is available for all subjects in the study cohort. Under the frame of marginal hazard model, we propose to estimate the induced relative risk function in the non-validation set through kernel smoothing method and then obtain an estimated pseudo-partial likelihood function. The proposed estimator which maximizes the estimated pseudo-partial likelihood is shown to be consistent and asymptotically normal. We also give an estimator of the marginal cumulative baseline hazard function. Simulation studies are conducted to evaluate the finite sample performance of our proposed estimator. The proposed method is illustrated by analyzing a heart disease data from the Study of Left Ventricular Dysfunction (SOLVD).     

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.     

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).     

On testing equality of pairwise rank correlations in a multivariate random vector

Spearman’s rank-correlation coefficient (also called Spearman’s rho) represents one of the best-known measures to quantify the degree of dependence between two random variables. As a copula-based dependence measure, it is invariant with respect to the distribution’s univariate marginal distribution functions. In this paper, we consider statistical tests for the hypothesis that all pairwise Spearman’s rank correlation coefficients in a multivariate random vector are equal. The tests are nonparametric and their asymptotic distributions are derived based on the asymptotic behavior of the empirical copula process. Only weak assumptions on the distribution function, such as continuity of the marginal distributions and continuous partial differentiability of the copula, are required for obtaining the results. A nonparametric bootstrap method is suggested for either estimating unknown parameters of the test statistics or for determining the associated critical values. We present a simulation study in order to investigate the power of the proposed tests. The results are compared to a classical parametric test for equal pairwise Pearson’s correlation coefficients in a multivariate random vector. The general setting also allows the derivation of a test for stochastic independence based on Spearman’s rho.     

Smooth estimation of survival and density functions for a stationary associated process using Poisson weights

Let {X_n,n≥1} be a sequence of stationary non-negative associated random variables with common marginal density f(x). Here we use the empirical survival function as studied in Bagai and Prakasa Rao (1991) and apply the smoothing technique proposed by Gawronski (1980) (see also Chaubey and Sen, 1996) in proposing a smooth estimator of the density function f and that of the corresponding survival function. Some asymptotic properties of the resulting estimators, similar to those obtained in Chaubey and Sen (1996) for the i.i.d. case, are derived. A simulation study has been carried out to compare the new estimator to the kernel estimator of a density function given in Bagai and Prakasa Rao (1996) and the estimator in Buch-Larsen et al. (2005).     

Likelihood-based kernel estimation in semiparametric errors-in-covariables models with validation data

We present methods to handle error-in-variables models. Kernel-based likelihood score estimating equation methods are developed for estimating conditional density parameters. In particular, a semiparametric likelihood method is proposed for sufficiently using the information in the data. The asymptotic distribution theory is derived. Small sample simulations and a real data set are used to illustrate the proposed estimation methods.     

Variance component testing in semiparametric mixed models

It is of considerable interest to test for heteroscedasticity in statistical studies. In this paper, we investigate such a problem under the framework of a semiparametric mixed model. A score test is proposed for the hypothesis that all the variance components are zero. We establish the asymptotic property of the test, and examine its performance in a simulation study. The test is illustrated with the analysis of a longitudinal study of measurements of serum creatinine.