Quasi-Minimax Estimation in the Linear Model with Measurement Errors

In this article, we consider quasi-minimax estimation in the linear regression model where some covariates are measured with additive errors. When measurement errors are directly ignored the minimax risk of the resulting estimator can be large. By correcting the attenuation we propose a penalized quadratic risk function. A simulation study is conducted to illustrate the performance of the proposed estimators.     

Adaptive Mixtures of Regressions: Improving Predictive Inference when Population has Changed

The present work investigates the estimation of regression mixtures when population has changed between the training and the prediction stages. Two approaches are proposed: a parametric approach modeling the relationship between dependent variables of both populations, and a Bayesian approach in which the priors on the prediction population depend on the mixture regression parameters of the training population. The relevance of both approaches is illustrated on simulations and on an environmental dataset.     

Local Linear Regression for Non Grid Spatiotemporal Models with Autoregressive Errors

In various environmental studies, spatiotemporal correlated data are involved, so there has been increasing demand of proposing spatiotemporal estimation methods that capture spatiotemporal correlation so as to improve the accuracy of estimation. In this article, we construct estimators for non grid spatiotemporal models with autoregressive errors. It is proved that the estimators are asymptotic normality. Simulation results also show the estimators perform well.     

Estimation of the mixtures of GLMs with covariate-dependent mixing proportions

ABSTRACT

In this article, we study the estimation for a class of semiparametric mixtures of generalized linear models where mixing proportions depend on a covariate non parametrically. We investigate a backfitting estimation procedure and show the asymptotic normality of the proposed estimators under mild conditions. We conduct simulation to show the good performance of our methodology and give a real data analysis as an illustration.     

Empirical Likelihood Inference of the Partial Linear Isotonic Errors-in-variables Regression Models with Missing Data

This article is concerned with statistical inference of the partial linear isotonic regression model missing response and measurement errors in covariates. We proposed an empirical likelihood ratio test statistics and show that it has a limiting weighted chi-square distribution. An adjusted empirical likelihood ratio statistic, which is shown to have a limiting standard central chi-square distribution, is then proposed further. A maximum empirical likelihood estimator is also developed. A simulation study is conducted to examine the finite-sample property of proposed procedure.     

Geometric View of Measurement Errors

The slope of the best fit line from minimizing the sum of the squared oblique errors is the root of a polynomial of degree four. This geometric view of measurement errors is used to give insight into the performance of various slope estimators for the measurement error model including an adjusted fourth moment estimator introduced by Gillard and Iles (2005 Gillard , J. , Iles , T. ( 2005 ). Method of moments estimation in linear regression with errors in both variables, Cardiff University School of Mathematics Technical Report, Cardiff, Wales, UK . [Google Scholar]) to remove the jump discontinuity in the estimator of Copas (1972 Copas , J. ( 1972 ). The likelihood surface in the linear functional relationship problem . Journal of the Royal Statistical Society. Series B (Methodological) 34 : 274 – 278 . [Google Scholar]). The polynomial of degree four is associated with a minimun deviation estimator. A simulation study compares these estimators showing improvement in bias and mean squared error.     

Mixtures of regression models with incomplete and noisy data

The estimation of the mixtures of regression models is usually based on the normal assumption of components and maximum likelihood estimation of the normal components is sensitive to noise, outliers, or high-leverage points. Missing values are inevitable in many situations and parameter estimates could be biased if the missing values are not handled properly. In this article, we propose the mixtures of regression models for contaminated incomplete heterogeneous data. The proposed models provide robust estimates of regression coefficients varying across latent subgroups even under the presence of missing values. The methodology is illustrated through simulation studies and a real data analysis.     

Kernel Density-Based Linear Regression Estimate

For linear regression models with non normally distributed errors, the least squares estimate (LSE) will lose some efficiency compared to the maximum likelihood estimate (MLE). In this article, we propose a kernel density-based regression estimate (KDRE) that is adaptive to the unknown error distribution. The key idea is to approximate the likelihood function by using a nonparametric kernel density estimate of the error density based on some initial parameter estimate. The proposed estimate is shown to be asymptotically as efficient as the oracle MLE which assumes the error density were known. In addition, we propose an EM type algorithm to maximize the estimated likelihood function and show that the KDRE can be considered as an iterated weighted least squares estimate, which provides us some insights on the adaptiveness of KDRE to the unknown error distribution. Our Monte Carlo simulation studies show that, while comparable to the traditional LSE for normal errors, the proposed estimation procedure can have substantial efficiency gain for non normal errors. Moreover, the efficiency gain can be achieved even for a small sample size.     

Linear Quantile Regression Based on EM Algorithm

This article aims to put forward a new method to solve the linear quantile regression problems based on EM algorithm using a location-scale mixture of the asymmetric Laplace error distribution. A closed form of the estimator of the unknown parameter vector β based on EM algorithm, is obtained. In addition, some simulations are conducted to illustrate the performance of the proposed method. Simulation results demonstrate that the proposed algorithm performs well. Finally, the classical Engel data is fitted and the Bootstrap confidence intervals for estimators are provided.     

Local Linear Kernel Regression with Long-Range Dependent Errors

This paper considers the use of a local linear kernel regression method to test whether the mean function of a sequence of long-range dependent processes has discontinuities or change-points. It proposes a non-parametric estimation procedure and then establishes an asymptotic theory for the estimation procedure. Examples, simulated and real, illustrate the estimation procedure.     

Likelihood-Based Inference for Multivariate Skew-Normal Regression Models

In this article, we present EM algorithms for performing maximum likelihood estimation for three multivariate skew-normal regression models of considerable practical interest. We also consider the restricted estimation of the parameters of certain important special cases of two models. The methodology developed is applied in the analysis of longitudinal data on dental plaque and cholesterol levels.     

Semiparametric Analysis of Isotonic Errors-in-Variables Regression Models with Missing Response

This article is concerned with the estimation problem in the semiparametric isotonic regression model when the covariates are measured with additive errors and the response is missing at random. An inverse marginal probability weighted imputation approach is developed to estimate the regression parameters and a least-square approach under monotone constraint is employed to estimate the functional component. We show that the proposed estimator of the regression parameter is root-n consistent and asymptotically normal and the isotonic estimator of the functional component, at a fixed point, is cubic root-n consistent. A simulation study is conducted to examine the finite-sample properties of the proposed estimators. A data set is used to demonstrate the proposed approach.     

A Semi-parametric Regression Model with Errors in Variables

Abstract.  In this paper, we consider a partial linear regression model with measurement errors in possibly all the variables. We use a method of moments and deconvolution to construct a new class of parametric estimators together with a non-parametric kernel estimator. Strong convergence, optimal rate of weak convergence and asymptotic normality of the estimators are investigated.     

Local Linear Additive Quantile Regression

Abstract.  We consider non-parametric additive quantile regression estimation by kernel-weighted local linear fitting. The estimator is based on localizing the characterization of quantile regression as the minimizer of the appropriate 'check function'. A backfitting algorithm and a heuristic rule for selecting the smoothing parameter are explored. We also study the estimation of average-derivative quantile regression under the additive model. The techniques are illustrated by a simulated example and a real data set.     

Instrument Assisted Regression for Errors in Variables Models with Binary Response

We study errors‐in‐variables problems when the response is binary and instrumental variables are available. We construct consistent estimators through taking advantage of the prediction relation between the unobservable variables and the instruments. The asymptotic properties of the new estimator are established and illustrated through simulation studies. We also demonstrate that the method can be readily generalized to generalized linear models and beyond. The usefulness of the method is illustrated through a real data example.     

Edgeworth Expansion for Linear Regression Processes with Long-Memory Errors

This article provides an Edgeworth expansion for the distribution of the log-likelihood derivative LLD of the parameter of a time series generated by a linear regression model with Gaussian, stationary, and long-memory errors. Under some sets of conditions on the regression coefficients, the spectral density function, and the parameter values, an Edgeworth expansion of the density as well as the distribution function of a vector of centered and normalized derivatives of the plug-in log-likelihood PLL function of arbitrarily large order is established. This is done by extending the results of Lieberman et al. (2003 Lieberman , O. , Rousseau , J. , Zucker , D. M. ( 2003 ). Valid edgeworth expansions for the maximum likelihood estimator of the parameter of a stationary. gaussian, strongly dependent processes. it Ann. Statist. 31:586–612 . [Google Scholar]), who provided an Edgeworth expansion for the Gaussian stationary long-memory case, to our present model, which is a linear regression process with stationary Gaussian long-memory errors.     

Finite Population Variance Estimation in Presence of Measurement Errors

In this article, we propose a class of estimators for the population variance of a quantity of interest. The estimators in the class use auxiliary information to improve efficiency, and we suppose that measurement errors are present both in the study and auxiliary variate. We take into account such problem using a regression approach. We show that the class proposed is quite flexible and general, allowing to consider many kinds of information as auxiliary one. Comparisons within estimators in the class are studied theoretically and through simulations.     

Asymptotic Properties of Marginal Least-Square Estimator for Ultrahigh-Dimensional Linear Regression Models with Correlated Errors

In this article, we discuss asymptotic properties of marginal least-square estimator for ultrahigh-dimensional linear regression models. We are specifically interested in probabilistic consistency of the marginal least-square estimator in the presence of correlated errors. We show that under a partial orthogonality condition, the marginal least-square estimator can achieve variable selection consistency. In addition, we demonstrate that if a mutual orthogonality holds, the marginal least-square estimator satisfies estimation consistency. The discussed theories are exemplified through extensive simulation studies.