Robust estimation for functional coefficient regression models with spatial data

A global smoothing procedure is developed using B-spline function approximation for estimating the unknown functions of a functional coefficient regression model with spatial data. A general formulation is used to treat mean regression, median regression, quantile regression and robust mean regression in one setting. The global convergence rates of the estimators of unknown coefficient functions are established. Various applications of the main results, including estimating conditional quantile coefficient functions and robustifying the mean regression coefficient functions are given. Finite sample properties of our procedures are studied through Monte Carlo simulations. A housing data example is used to illustrate the proposed methodology.     

Partially functional linear varying coefficient model

In this paper, we introduce a new partially functional linear varying coefficient model, where the response is a scalar and some of the covariates are functional. By means of functional principal components analysis and local linear smoothing techniques, we obtain the estimators of coefficient functions of both function-valued variable and real-valued variables. Then the rates of convergence of the proposed estimators and the mean squared prediction error are established under some regularity conditions. Moreover, we develop a hypothesis test for the model and employ the bootstrap procedure to evaluate the null distribution of test statistic and the p-value of the test. At last, we illustrate the finite sample performance of our methods with some simulation studies and a real data application.     

Modified SEE variable selection for varying coefficient instrumental variable models

We consider the problem of variable selection for a class of varying coefficient models with instrumental variables. We focus on the case that some covariates are endogenous variables, and some auxiliary instrumental variables are available. An instrumental variable based variable selection procedure is proposed by using modified smooth-threshold estimating equations (SEEs). The proposed procedure can automatically eliminate the irrelevant covariates by setting the corresponding coefficient functions as zero, and simultaneously estimate the nonzero regression coefficients by solving the smooth-threshold estimating equations. The proposed variable selection procedure avoids the convex optimization problem, and is flexible and easy to implement. Simulation studies are carried out to assess the performance of the proposed variable selection method.     

Simultaneous Confidence Bands and Hypothesis Testing in Varying-coefficient Models

Regression analysis is one of the most commonly used techniques in statistics. When the dimension of independent variables is high, it is difficult to conduct efficient non-parametric analysis straightforwardly from the data. As an important alternative to the additive and other non-parametric models, varying-coefficient models can reduce the modelling bias and avoid the "curse of dimensionality" significantly. In addition, the coefficient functions can easily be estimated via a simple local regression. Based on local polynomial techniques, we provide the asymptotic distribution for the maximum of the normalized deviations of the estimated coefficient functions away from the true coefficient functions. Using this result and the pre-asymptotic substitution idea for estimating biases and variances, simultaneous confidence bands for the underlying coefficient functions are constructed. An important question in the varying coefficient models is whether an estimated coefficient function is statistically significantly different from zero or a constant. Based on newly derived asymptotic theory, a formal procedure is proposed for testing whether a particular parametric form fits a given data set. Simulated and real-data examples are used to illustrate our techniques.     

Statistical Inference and Applications of Mixture of Varying Coefficient Models

In this paper, we consider a new mixture of varying coefficient models, in which each mixture component follows a varying coefficient model and the mixing proportions and dispersion parameters are also allowed to be unknown smooth functions. We systematically study the identifiability, estimation and inference for the new mixture model. The proposed new mixture model is rather general, encompassing many mixture models as its special cases such as mixtures of linear regression models, mixtures of generalized linear models, mixtures of partially linear models and mixtures of generalized additive models, some of which are new mixture models by themselves and have not been investigated before. The new mixture of varying coefficient model is shown to be identifiable under mild conditions. We develop a local likelihood procedure and a modified expectation–maximization algorithm for the estimation of the unknown non‐parametric functions. Asymptotic normality is established for the proposed estimator. A generalized likelihood ratio test is further developed for testing whether some of the unknown functions are constants. We derive the asymptotic distribution of the proposed generalized likelihood ratio test statistics and prove that the Wilks phenomenon holds. The proposed methodology is illustrated by Monte Carlo simulations and an analysis of a CO₂‐GDP data set.     

Regression analysis of multivariate current status data under a varying coefficients additive hazards frailty model

This article discusses regression analysis of multivariate current status failure time data for which the observation time may be related to the underlying survival time. A local partial likelihood technique is used to estimate the varying coefficient covariate effect functions under the additive hazards frailty model. The asymptotic properties of the proposed estimators are established. An extensive simulation study is conducted for the evaluation of the proposed procedure, the results of which indicate that the proposed method works well in practice. Also, a real data study is provided to illustrate the performance of the proposed method.     

Quantile regression for robust inference on varying coefficient partially nonlinear models

In this paper, we propose a robust statistical inference approach for the varying coefficient partially nonlinear models based on quantile regression. A three-stage estimation procedure is developed to estimate the parameter and coefficient functions involved in the model. Under some mild regularity conditions, the asymptotic properties of the resulted estimators are established. Some simulation studies are conducted to evaluate the finite performance as well as the robustness of our proposed quantile regression method versus the well known profile least squares estimation procedure. Moreover, the Boston housing price data is given to further illustrate the application of the new method.     

Semiparametric Time-Varying Coefficients Regression Model for Longitudinal Data

Abstract.  In this paper, we consider a semiparametric time-varying coefficients regression model where the influences of some covariates vary non-parametrically with time while the effects of the remaining covariates follow certain parametric functions of time. The weighted least squares type estimators for the unknown parameters of the parametric coefficient functions as well as the estimators for the non-parametric coefficient functions are developed. We show that the kernel smoothing that avoids modelling of the sampling times is asymptotically more efficient than a single nearest neighbour smoothing that depends on the estimation of the sampling model. The asymptotic optimal bandwidth is also derived. A hypothesis testing procedure is proposed to test whether some covariate effects follow certain parametric forms. Simulation studies are conducted to compare the finite sample performances of the kernel neighbourhood smoothing and the single nearest neighbour smoothing and to check the empirical sizes and powers of the proposed testing procedures. An application to a data set from an AIDS clinical trial study is provided for illustration.     

Robust variable selection in modal varying-coefficient models with longitudinal

In this article we present a robust and efficient variable selection procedure by using modal regression for varying-coefficient models with longitudinal data. The new method is proposed based on basis function approximations and a group version of the adaptive LASSO penalty, which can select significant variables and estimate the non-zero smooth coefficient functions simultaneously. Under suitable conditions, we establish the consistency in variable selection and the oracle property in estimation. A simulation study and two real data examples are undertaken to assess the finite sample performance of the proposed variable selection procedure.     

Two-dimensional Renewal Function Approximation

Two-dimensional renewal functions, which are naturally extensions of one-dimensional renewal functions, have wide applicability in areas where two random variables are needed to characterize the underlying process. These functions satisfy the renewal equation, which is not amenable for analytical solutions. This paper proposes a simple approximation for the computation of the two- dimensional renewal function based only on the first two moments and the correlation coefficient of the variables. The approximation yields exact values of renewal function for bivariate exponential distribution function. Illustrations are presented to compare our approximation with that of Iskandar (1991) who provided a computational procedure which requires the use of the bivariate distribution function of the two variables. A two-dimensional warranty model is used to illustrate the approximation.     

Wavelet estimation in time-varying coefficient time series models with measurement errors

The article studies a time-varying coefficient time series model in which some of the covariates are measured with additive errors. In order to overcome the bias of estimator of the coefficient functions when measurement errors are ignored, we propose a modified least squares estimator based on wavelet procedures. The advantage of the wavelet method is to avoid the restrictive smoothness requirement for varying-coefficient functions of the traditional smoothing approaches, such as kernel and local polynomial methods. The asymptotic properties of the proposed wavelet estimators are established under the α-mixing conditions and without specifying the error distribution. These results can be used to make asymptotically valid statistical inference.     

A test for zero point triserial correlation coefficient and its relationship with tests for homogeneity of means with ordered alternatives

The point triserial correlation coefficient is defined and, under appropriate order restrictions, an exact test that this correlation coefficient equals zero is developed. The power function of that test is derived and partially tabulated. The general problem of testing for homogeneity of means under ordered alternatives is discussed. The available procedures for performing such tests are considered, are seen to provide alternative approaches to the test developed herein, and are compared with that test. An exact test for the equality of dependent point triserial correlation coefficients is described through application of a procedure suggested by Wolfe ‘1976’     

零膨胀计数数据函数型部分变系数模型

在实际数据分析中经常会遇到零膨胀计数数据作为响应变量与函数型随机变量和随机向量作为预测变量相关联。本文考虑函数型部分变系数零膨胀模型 (FPVCZIM),模型中无穷维的斜率函数用函数型主成分基逼近,系数函数用B-样条进行拟合。通过EM 算法得到估计量,讨论其理论性质,在一些正则条件下获得了斜率函数和系数函数估计量的收敛速度。有限样本的Monte Carlo 模拟研究和真实数据分析被用来解释本文提出的方法。     

Bayesian and maximum likelihood estimations of the inverted exponentiated half logistic distribution under progressive Type II censoring

In this paper, the estimation of parameters, reliability and hazard functions of a inverted exponentiated half logistic distribution (IEHLD) from progressive Type II censored data has been considered. The Bayes estimates for progressive Type II censored IEHLD under asymmetric and symmetric loss functions such as squared error, general entropy and linex loss function are provided. The Bayes estimates for progressive Type II censored IEHLD parameters, reliability and hazard functions are also obtained under the balanced loss functions. However, the Bayes estimates cannot be obtained explicitly, Lindley approximation method and importance sampling procedure are considered to obtain the Bayes estimates. Furthermore, the asymptotic normality of the maximum likelihood estimates is used to obtain the approximate confidence intervals. The highest posterior density credible intervals of the parameters based on importance sampling procedure are computed. Simulations are performed to see the performance of the proposed estimates. For illustrative purposes, two data sets have been analyzed.     

Weighted composite quantile regression for partially linear varying coefficient models

Partially linear varying coefficient models (PLVCMs) with heteroscedasticity are considered in this article. Based on composite quantile regression, we develop a weighted composite quantile regression (WCQR) to estimate the non parametric varying coefficient functions and the parametric regression coefficients. The WCQR is augmented using a data-driven weighting scheme. Moreover, the asymptotic normality of proposed estimators for both the parametric and non parametric parts are studied explicitly. In addition, by comparing the asymptotic relative efficiency theoretically and numerically, WCQR method all outperforms the CQR method and some other estimate methods. To achieve sparsity with high-dimensional covariates, we develop a variable selection procedure to select significant parametric components for the PLVCM and prove the method possessing the oracle property. Both simulations and data analysis are conducted to illustrate the finite-sample performance of the proposed methods.     

Variable selection for semiparametric varying coefficient partially linear model based on modal regression with missing data

Abstract

In this article, we focus on the variable selection for semiparametric varying coefficient partially linear model with response missing at random. Variable selection is proposed based on modal regression, where the non parametric functions are approximated by B-spline basis. The proposed procedure uses SCAD penalty to realize variable selection of parametric and nonparametric components simultaneously. Furthermore, we establish the consistency, the sparse property and asymptotic normality of the resulting estimators. The penalty estimation parameters value of the proposed method is calculated by EM algorithm. Simulation studies are carried out to assess the finite sample performance of the proposed variable selection procedure.     

Statistical inference on partial linear additive models with distortion measurement errors

We consider statistical inference for partial linear additive models (PLAMs) when the linear covariates are measured with errors and distorted by unknown functions of commonly observable confounding variables. A semiparametric profile least squares estimation procedure is proposed to estimate unknown parameter under unrestricted and restricted conditions. Asymptotic properties for the estimators are established. To test a hypothesis on the parametric components, a test statistic based on the difference between the residual sums of squares under the null and alternative hypotheses is proposed, and we further show that its limiting distribution is a weighted sum of independent standard chi-squared distributions. A bootstrap procedure is further proposed to calculate critical values. Simulation studies are conducted to demonstrate the performance of the proposed procedure and a real example is analyzed for an illustration.     

Multi-treatment covariate-adjusted adaptive design under informative censoring

In the present work, we find a set of reliability functionals to fix up an allocation strategy among K(≥2) treatments when the response distributions, conditionally dependent on some continuous prognostic variable, are exponential with unknown linear regression functions as the means of the respective conditional distributions. Targeting such reliability functionals, we propose a covariate-adjusted response-adaptive randomization procedure for the multi-treatment single-period clinical trial under the Koziol–Green model for informative censoring. We compare the proposed procedure with its competitive covariate-eliminated procedure.     

Weighted quantile regression and testing for varying-coefficient models with randomly truncated data

This paper develops a varying-coefficient approach to the estimation and testing of regression quantiles under randomly truncated data. In order to handle the truncated data, the random weights are introduced and the weighted quantile regression (WQR) estimators for nonparametric functions are proposed. To achieve nice efficiency properties, we further develop a weighted composite quantile regression (WCQR) estimation method for nonparametric functions in varying-coefficient models. The asymptotic properties both for the proposed WQR and WCQR estimators are established. In addition, we propose a novel bootstrap-based test procedure to test whether the nonparametric functions in varying-coefficient quantile models can be specified by some function forms. The performance of the proposed estimators and test procedure are investigated through simulation studies and a real data example.     

Asymptotic Normality of M-Estimators for Varying Coefficient Models with Longitudinal Data

This article considers a nonparametric varying coefficient regression model with longitudinal observations. The relationship between the dependent variable and the covariates is assumed to be linear at a specific time point, but the coefficients are allowed to change over time. A general formulation is used to treat mean regression, median regression, quantile regression, and robust mean regression in one setting. The local M-estimators of the unknown coefficient functions are obtained by local linear method. The asymptotic distributions of M-estimators of unknown coefficient functions at both interior and boundary points are established. Various applications of the main results, including estimating conditional quantile coefficient functions and robustifying the mean regression coefficient functions are derived. Finite sample properties of our procedures are studied through Monte Carlo simulations.