Varying Coefficient Functions Research Articles

Functional Coefficient Nonstationary Regression

This paper studies a general class of nonlinear varying coefficient time series models with possible nonstationarity in both the regressors and the varying coffiecient components. The model accommodates a cointegrating structure and allows for endogeneity with contemporaneous correlation among the regressors, the varying coefficient drivers, and the residuals. This framework allows for a mixture of stationary and non-stationary data and is well suited to a variety of models that are commonly used in applied econometric work. Nonparametric and semiparametric estimation methods are proposed to estimate the varying coefficient functions. The analytical findings reveal some important differences, including convergence rates, that can arise in the conduct of semiparametric regression with nonstationary data. The results include some new asymptotic theory for nonlinear functionals of nonstationary and stationary time series that are of wider interest and applicability and subsume much earlier research on such systems. The finite sample properties of the proposed econometric methods are analyzed in simulations. An empirical illustration examines nonlinear dependencies in aggregate consumption function behavior in the US over the period 1960-2009.

Simplified error bounds for turning point expansions

Recently, the present authors derived new asymptotic expansions for linear differential equations having a simple turning point. These involve Airy functions and slowly varying coefficient functions, and were simpler than previous approximations, in particular being computable to a high degree of accuracy. Here we present explicit error bounds for these expansions which only involve elementary functions, and thereby provide a simplification of the bounds associated with the classical expansions of Olver.

Efficient Estimation for Varying-Coefficient Mixed Effects Models with Functional Response Data

In this article, we focus on the estimation of varying-coefficient mixed effects models for longitudinal and sparse functional response data, by using the generalized least squares method coupling a modified local kernel smoothing technique. This approach provides a useful framework that simultaneously takes into account the within-subject covariance and all observation information in the estimation to improve efficiency. We establish both uniform consistency and pointwise asymptotic normality for the proposed estimators of varying-coefficient functions. Numerical studies are carried out to illustrate the finite sample performance of the proposed procedure. An application to the white matter tract dataset obtained from Alzheimer’s Disease Neuroimaging Initiative study is also provided.

Sparse Learning and Structure Identification for Ultrahigh-Dimensional Image-on-Scalar Regression

This article considers high-dimensional image-on-scalar regression, where the spatial heterogeneity of covariate effects on imaging responses is investigated via a flexible partially linear spatially varying coefficient model. To tackle the challenges of spatial smoothing over the imaging response’s complex domain consisting of regions of interest, we approximate the spatially varying coefficient functions via bivariate spline functions over triangulation. We first study estimation when the active constant coefficients and varying coefficient functions are known in advance. We then further develop a unified approach for simultaneous sparse learning and model structure identification in the presence of ultrahigh-dimensional covariates. Our method can identify zero, nonzero constant, and spatially varying components correctly and efficiently. The estimators of constant coefficients and varying coefficient functions are consistent and asymptotically normal for constant coefficient estimators. The method is evaluated by Monte Carlo simulation studies and applied to a dataset provided by the Alzheimer’s Disease Neuroimaging Initiative. Supplementary materials for this article are available online.

Empirical likelihood in varying-coefficient quantile regression with missing observations

In this paper, we focus on the partially linear varying-coefficient quantile regression model with observations missing at random (MAR), which include the responses or the responses and covariates MAR. Based on the local linear estimation of the varying-coefficient function in the model, we construct empirical log-likelihood ratio functions for unknown parameter in the linear part of the model, which are proved to be asymptotically weighted chi-squared distributions, further the adjusted empirical log-likelihood ratio functions are verified to converge to standard chi-squared distribution. The asymptotic normality of maximum empirical likelihood estimator for the parameter is also established. In order to do variable selection, we consider penalized empirical likelihood by using smoothly clipped absolute deviationv (SCAD) penalty, and the oracle property of the penalized likelihood estimator of the parameter is proved. Furthermore, Monte Carlo simulation and a real data analysis are undertaken to test the performance of the proposed methods.

Spatial autoregressive partially linear varying coefficient models

ABSTRACTIn this article, we consider a class of partially linear spatially varying coefficient autoregressive models for data distributed over complex domains. We propose approximating the varying coefficient functions via bivariate splines over triangulation to deal with the complex boundary of the spatial domain. Under some regularity conditions, the estimated constant coefficients are asymptotically normally distributed, and the estimated varying coefficients are consistent and possess the optimal convergence rate. A penalized bivariate spline estimation method with a more flexible choice of triangulation is proposed. We further develop a fast algorithm to calculate the geodesic distance. The proposed method is much more computationally efficient than the local smoothing methods, and thus capable of handling large scales of spatial data. In addition, we propose a model selection approach to identify predictors with constant and varying effects. The performance of the proposed method is evaluated by simulation examples and the Sydney real estate dataset.

Nonparametric covariance estimation with shrinkage toward stationary models

AbstractEstimation of an unstructured covariance matrix is difficult because of the challenges posed by parameter space dimensionality and the positive‐definiteness constraint that estimates should satisfy. We consider a general nonparametric covariance estimation framework for longitudinal data using the Cholesky decomposition of a positive‐definite matrix. The covariance matrix of time‐ordered measurements is diagonalized by a lower triangular matrix with unconstrained entries that are statistically interpretable as parameters for a varying coefficient autoregressive model. Using this dual interpretation of the Cholesky decomposition and allowing for irregular sampling time points, we treat covariance estimation as bivariate smoothing and cast it in a regularization framework for desired forms of simplicity in covariance models. Viewing stationarity as a form of simplicity or parsimony in covariance, we model the varying coefficient function with components depending on time lag and its orthogonal direction separately and penalize the components that capture the nonstationarity in the fitted function. We demonstrate construction of a covariance estimator using the smoothing spline framework. Simulation studies establish the advantage of our approach over alternative estimators proposed in the longitudinal data setting. We analyze a longitudinal dataset to illustrate application of the methodology and compare our estimates to those resulting from alternative models.This article is categorized under: Data: Types and Structure > Time Series, Stochastic Processes, and Functional Data Statistical and Graphical Methods of Data Analysis > Nonparametric Methods Algorithms and Computational Methods > Maximum Likelihood Methods

Estimation of semi-varying coefficient error-in-variable models with surrogate data and validation sample

In this study, a semi-varying coefficient error-in-variable model with surrogate data and validation sample is proposed. Without specifying any error structure, we firstly use the local linear kernel smoothing technique to define the estimators and the proposed estimators are proved to be asymptotically normal. Then, we conduct generalized likelihood ratio (GLR) test on varying coefficient function. The data–driven bandwidth selection method is discussed. Finally, simulated studies are conducted to illustrate the finite sample properties of the proposed estimators and efficiency of the GLR methodology.

A multilevel mixed effects varying coefficient model with multilevel predictors and random effects for modeling hospitalization risk in patients on dialysis.

For patients on dialysis, hospitalizations remain a major risk factor for mortality and morbidity. We use data from a large national database, United States Renal Data System, to model time-varying effects of hospitalization risk factors as functions of time since initiation of dialysis. To account for the three-level hierarchical structure in the data where hospitalizations are nested in patients and patients are nested in dialysis facilities, we propose a multilevel mixed effects varying coefficient model (MME-VCM) where multilevel (patient- and facility-level) random effects are used to model the dependence structure of the data. The proposed MME-VCM also includes multilevel covariates, where baseline demographics and comorbidities are among the patient-level factors, and staffing composition and facility size are among the facility-level risk factors. To address the challenge of high-dimensional integrals due to the hierarchical structure of the random effects, we propose a novel two-step approximate EM algorithm based on the fully exponential Laplace approximation. Inference for the varying coefficient functions and variance components is achieved via derivation of the standard errors using score contributions. The finite sample performance of the proposed estimation procedure is studied throughsimulations.

Efficient estimation of a varying-coefficient partially linear proportional hazards model with current status data

ABSTRACTWe consider a varying-coefficient partially linear proportional hazards model with current status data. The proposed model enables one to examine the extent to which some covariates interact nonlinearly with an exposure variable, while other covariates present linear effects. B-splines are applied to model both the unknown cumulative baseline hazard function and the varying-coefficient functions with and without monotone constraints, depending on the nature of the nonparametric functions. The sieve maximum likelihood estimation method is used to get an integrated estimate for the linear coefficients, the varying-coefficient functions and the cumulative baseline hazard function. The proposed parameter estimators are proved to be semiparametrically efficient and asymptotically normal, and the estimators for the nonparametric functions achieve the optimal rate of convergence. Simulation studies and a real data analysis are used for assessment and illustration.

A New Functional Estimation Procedure for Varying Coefficient Models

There has been substantial research interest in developing various estimation procedures for varying coefficient models. Most methods in the literature require specifying a working covariance structure. In case of a misspecified structure, estimation of the varying coefficient function may be deficient. Taking advantage of functional principal component analysis, we propose a new functional estimation procedure for varying coefficient models which does not need a working covariance structure. Weak convergence property for the proposed estimators has been established. Based on the simulation studies, the proposed procedure works better than the naive local linear regression with working independence error structure by Zhu et al. and Cholesky decomposition method by Lin et al. We apply our method to analyze the growth data of newborn infants in a real medical study and produce interpretable results.

Linear regression models with general distortion measurement errors

This article considers linear regression models when neither the response variable nor the covariates can be directly observed, but are measured with both multiplicative and additive distortion measurement errors. We transform the linear regression models via the varying coefficient models, then moment-based estimators are proposed by using the estimated varying coefficient functions. We study the asymptotic results of the proposed estimator, and construct a test statistic to check whether the coefficient in the linear model is zero or not. Lastly, we make some comparisons between the proposed estimators and other existing estimators through the simulation.

Finite mixture of varying coefficient model: Estimation and component selection

Heterogeneous longitudinal data have become prevalent in medical, biological, and social studies. This paper proposes a finite mixture of varying coefficient models for handling heterogeneous populations. Each component of the mixture is modeled by a varying coefficient mixed-effect model that characterizes the longitudinal relations among variables. The identifiability of the mixture model is studied. Regression splines with equally spaced knots are used to approximate the varying coefficient functions, and a nested expectation maximization algorithm is developed to obtain the maximum likelihood estimation. We propose a penalized likelihood method based on the smoothly clipped absolute deviation (SCAD) penalty for the component selection of finite mixture of varying coefficient model. A modified BIC-based criterion based on the SCAD penalty, the BICSCAD, is proposed for selecting the penalty parameter and spline space simultaneously. The asymptotic properties of parameter estimation and component selection consistency are studied under mild conditions. Simulation studies are conducted to illustrate the component selection, parameter estimation, and inference of the proposed method. The model is then applied to a heterogeneous longitudinal data set from a study of the treatment effect on the use of heroin in the California Civil Addict Program.

Adaptive GDSW Coarse Spaces for Overlapping Schwarz Methods in Three Dimensions

A robust two-level overlapping Schwarz method for scalar elliptic model problems with highly varying coefficient functions is introduced. While the convergence of standard coarse spaces may depend ...

FMEM: Functional Mixed Effects Models for Longitudinal Functional Responses.

The aim of this paper is to conduct a systematic and theoretical analysis of estimation and inference for a class of functional mixed effects models (FMEM). Such FMEMs consist of fixed effects that characterize the association between longitudinal functional responses and covariates of interest and random effects that capture the spatial-temporal correlations of longitudinal functional responses. We propose local linear estimates of refined fixed effect functions and establish their weak convergence along with a simultaneous confidence band for each fixed-effect function. We propose a global test for the linear hypotheses of varying coefficient functions and derive the associated asymptotic distribution under the null hypothesis and the asymptotic power under the alternative hypothesis are derived. We also establish the convergence rates of the estimated spatial-temporal covariance operators and their associated eigenvalues and eigenfunctions. We conduct extensive simulations and apply our method to a white-matter fiber data set from a national database for autism research to examine the finite-sample performance of the proposed estimation and inference procedures.

Efficient estimation of the varying-coefficient partially linear proportional odds model with current status data

We consider a varying-coefficient partially linear proportional odds model with current status data. This model enables one to examine the extent to which some covariates interact nonlinearly with an exposure variable, while other covariates present linear effects. B-spline approach and sieve maximum likelihood estimation method are used to get an integrated estimate for the linear coefficients, the varying-coefficient functions and the baseline function. The proposed parameter estimators are proved to be semiparametrically efficient and asymptotically normal, and the estimators for the nonparametric functions achieve the optimal rate of convergence. Simulation studies and a real data analysis are used for assessment and illustration.

Smoothed empirical likelihood inference via the modified Cholesky decomposition for quantile varying coefficient models with longitudinal data

It is essential to deal with the within-subject correlation among repeated measures over time to improve statistical inference efficiency. However, it is a challenging task to correctly specify a working correlation in quantile regression with longitudinal data. In this paper, we first develop an adaptive approach to estimate the within-subject covariance matrix of quantile regression by applying a modified Cholesky decomposition. Then, weighted kernel GEE-type quantile estimating equations are proposed for varying coefficient functions. Note that the proposed estimating equations include a discrete indicator function, which results in some problems for computation and asymptotic analysis. Thus, we construct smoothed estimating equations by introducing a bounded kernel function. Furthermore, we develop a smoothed empirical likelihood method to improve the accuracy of interval estimation. Finally, simulation studies and a real data analysis indicate that the proposed method has superior advantages over the existing methods in terms of coverage accuracies and widths of confidence intervals.

Modeling time-varying effects of multilevel risk factors of hospitalizations in patients on dialysis.

For chronic dialysis patients, a unique population requiring continuous medical care, methodologies to monitor patient outcomes, such as hospitalizations, over time, after initiation of dialysis, are of particular interest. Contributing to patient hospitalizations is a number of multilevel covariates such as demographics and comorbidities at the patient level and staffing composition at the dialysis facility level. We propose a varying coefficient model for multilevel risk factors (VCM-MR) to study the time-varying effects of covariates on patient hospitalization risk as a function of time on dialysis. The proposed VCM-MR also includes subject-specific random effects to account for within-subject correlation and dialysis facility-specific fixed effect varying coefficient functions to allow for the modeling of flexible time-varying facility-specific risk trajectories. An approximate EM algorithm and an iterative Newton-Raphson approach are proposed to address the challenge of estimation of high-dimensional parameters (varying coefficient functions) for thousands of dialysis facilities in the United States. The proposed modeling allows for comparisons between time-varying effects of multilevel risk factors as well as testing of facility-specific fixed effects. The method is applied to model hospitalization risk using the rich hierarchical data available on dialysis patients initiating dialysis between January 1, 2006 and December 31, 2008 from the United States Renal Data System, a large national database, where 331 443 hospitalizations over time are nested within patients, and 89 889 patients are nested within 2201 dialysis facilities. Patients are followed-up until December 31, 2013, where the follow-up time is truncated five years after the initiation of dialysis. Finite sample properties are studied through extensive simulations.

Semiparametric quantile estimation for varying coefficient partially linear measurement errors models

We study varying coefficient partially linear models when some linear covariates are error-prone, but their ancillary variables are available. After calibrating the error-prone covariates, we study quantile regression estimates for parametric coefficients and nonparametric varying coefficient functions, and we develop a semiparametric composite quantile estimation procedure. Asymptotic properties of the proposed estimators are established, and the estimators achieve their best convergence rate with proper bandwidth conditions. Simulation studies are conducted to evaluate the performance of the proposed method, and a real data set is analyzed as an illustration.

FSEM: Functional Structural Equation Models for Twin Functional Data

ABSTRACTThe aim of this article is to develop a novel class of functional structural equation models (FSEMs) for dissecting functional genetic and environmental effects on twin functional data, while characterizing the varying association between functional data and covariates of interest. We propose a three-stage estimation procedure to estimate varying coefficient functions for various covariates (e.g., gender) as well as three covariance operators for the genetic and environmental effects. We develop an inference procedure based on weighted likelihood ratio statistics to test the genetic/environmental effect at either a fixed location or a compact region. We also systematically carry out the theoretical analysis of the estimated varying functions, the weighted likelihood ratio statistics, and the estimated covariance operators. We conduct extensive Monte Carlo simulations to examine the finite-sample performance of the estimation and inference procedures. We apply the proposed FSEM to quantify the degree of genetic and environmental effects on twin white matter tracts obtained from the UNC early brain development study. Supplementary materials for this article are available online.