Robust estimation of multivariate regression model

This paper studies robust estimation of multivariate regression model using kernel weighted local linear regression. A robust estimation procedure is proposed for estimating the regression function and its partial derivatives. The proposed estimators are jointly asymptotically normal and attain nonparametric optimal convergence rate. One-step approximations to the robust estimators are introduced to reduce computational burden. The one-step local M-estimators are shown to achieve the same efficiency as the fully iterative local M-estimators as long as the initial estimators are good enough. The proposed estimators inherit the excellent edge-effect behavior of the local polynomial methods in the univariate case and at the same time overcome the disadvantages of the local least-squares based smoothers. Simulations are conducted to demonstrate the performance of the proposed estimators. Real data sets are analyzed to illustrate the practical utility of the proposed methodology. This work was supported by the National Natural Science Foundation of China (Grant No. 10471006).     

On nonparametric regression estimators based on regression quantiles

In the ciassical regression model Y_i=h(x_i) + ? _i, i=1,…,n, Cheng (1984) introduced linear combinations of regression quantiles as a new class of estimators for the unknown regression function h(x). The asymptotic properties studied in Cheng (1984) are reconsidered. We obtain a sharper scrong consistency rate and we improve on the conditions for asymptotic normality by proving a new result on the remainder term in the Bahadur representation for regression quantiles.     

A two-way classification of regression estimation strategies in probability sampling

This paper examines strategies for estimating the mean of a finite population in the following situation: A linear regression model is assumed to describe the population scatter. Various estimators β for the vector of regression parameters β are considered. Several ways of transforming each estimator β into a model-based estimator for the population mean are considered. Some estimators constructed in this way become sensitive to correctness of the assumed model. The estimators favoured in this paper are the ones in which the observations are weighted to reflect the sampling design, so that asymptotic design unbiasedness is achieved. For these estimators, the randomization distribution gives protection against model breakdown.     

Dimension reduction via local rank regression

Outer product of gradients (OPG) achieves dimension reduction via estimating the gradients of the regression function. In this paper, we propose two novel OPG estimators via local rank regression: the rank OPG estimator and the Walsh-average OPG estimator. Both proposals guard against a wide range of error distributions, and are safe alternatives to existing OPG estimators based on local linear regression or local L1 regression. The effectiveness of the new proposals are demonstrated via extensive numerical studies.     

A new modified Jackknifed estimator for the Poisson regression model

The Poisson regression is very popular in applied researches when analyzing the count data. However, multicollinearity problem arises for the Poisson regression model when the independent variables are highly intercorrelated. Shrinkage estimator is a commonly applied solution to the general problem caused by multicollinearity. Recently, the ridge regression (RR) estimators and some methods for estimating the ridge parameter k in the Poisson regression have been proposed. It has been found that some estimators are better than the commonly used maximum-likelihood (ML) estimator and some other RR estimators. In this study, the modified Jackknifed Poisson ridge regression (MJPR) estimator is proposed to remedy the multicollinearity. A simulation study and a real data example are provided to evaluate the performance of estimators. Both mean-squared error and the percentage relative error are considered as the performance criteria. The simulation study and the real data example results show that the proposed MJPR method outperforms the Poisson ridge regression, Jackknifed Poisson ridge regression and the ML in all of the different situations evaluated in this paper.     

A Jackknifed estimators for the negative binomial regression model

Shrinkage estimator is a commonly applied solution to the general problem caused by multicollinearity. Recently, the ridge regression (RR) estimators for estimating the ridge parameter k in the negative binomial (NB) regression have been proposed. The Jackknifed estimators are obtained to remedy the multicollinearity and reduce the bias. A simulation study is provided to evaluate the performance of estimators. Both mean squared error (MSE) and the percentage relative error (PRE) are considered as the performance criteria. The simulated result indicated that some of proposed Jackknifed estimators should be preferred to the ML method and ridge estimators to reduce MSE and bias.     

Partial covariate adjusted regression

Covariate adjusted regression (CAR) is a recently proposed adjustment method for regression analysis where both the response and predictors are not directly observed [?entürk, D., Müller, H.G., 2005. Covariate adjusted regression. Biometrika 92, 75–89]. The available data have been distorted by unknown functions of an observable confounding covariate. CAR provides consistent estimators for the coefficients of the regression between the variables of interest, adjusted for the confounder. We develop a broader class of partial covariate adjusted regression (PCAR) models to accommodate both distorted and undistorted (adjusted/unadjusted) predictors. The PCAR model allows for unadjusted predictors, such as age, gender and demographic variables, which are common in the analysis of biomedical and epidemiological data. The available estimation and inference procedures for CAR are shown to be invalid for the proposed PCAR model. We propose new estimators and develop new inference tools for the more general PCAR setting. In particular, we establish the asymptotic normality of the proposed estimators and propose consistent estimators of their asymptotic variances. Finite sample properties of the proposed estimators are investigated using simulation studies and the method is also illustrated with a Pima Indians diabetes data set.     

Superiority comparisons between mixed regression estimators

This paper gives necessary and sufficient conditions for a mixed regression estimator to be superior to another mixed estimator. The comparisons are based on the mean square error matrices of the estimators. Both estimators are allowed to be biased.     

Fractional principal components regression: a general approach to biased estimators

Several biased estimators have been proposed as alternatives to the least squares estimator when multicollinearity is present in the multiple linear regression model. The ridge estimator and the principal components estimator are two techniques that have been proposed for such problems. In this paper the class of fractional principal component estimators is developed for the multiple linear regression model. This class contains many of the biased estimators commonly used to combat multicollinearity. In the fractional principal components framework, two new estimation techniques are introduced. The theoretical performances of the new estimators are evaluated and their small sample properties are compared via simulation with the ridge, generalized ridge and principal components estimators     

Weighted nonparametric regression

In the fixed design regression model, additional weights are considered for the Nad a ray a-Watson and Gasser-Miiller kernel estimators. We study their asymptotic behavior and the relationships between new and classical estimators. For a simple family of weights, and considering the AIMSEAS global loss criterion, we show some possible theoretical advantages. An empirical study illustrates the performance of the weighted kernel estimators in theoretical ideal situations and in simulated data sets. Also some results concerning the use of weights for local polynomial estimators are given.     

Bridge regression: Adaptivity and group selection

In high-dimensional regression problems regularization methods have been a popular choice to address variable selection and multicollinearity. In this paper we study bridge regression that adaptively selects the penalty order from data and produces flexible solutions in various settings. We implement bridge regression based on the local linear and quadratic approximations to circumvent the nonconvex optimization problem. Our numerical study shows that the proposed bridge estimators are a robust choice in various circumstances compared to other penalized regression methods such as the ridge, lasso, and elastic net. In addition, we propose group bridge estimators that select grouped variables and study their asymptotic properties when the number of covariates increases along with the sample size. These estimators are also applied to varying-coefficient models. Numerical examples show superior performances of the proposed group bridge estimators in comparisons with other existing methods.     

Consistent estimation of regression parameters under replicated ultrastructural model with non-normal errors

This article discusses the construction and efficiency properties of consistent estimators of regression parameters under replicated ultrastructural model with not necessarily normally distributed measurement errors. The variances of measurement errors associated with the study and explanatory variables are estimated from the replicated sample observations and are used for the consistent estimation of regression parameters. The asymptotic efficiency properties of the estimators are derived and analysed. The finite sample performance of the estimators is empirically studied through a Monte Carlo simulation.     

Semiparametric log-linear regression for longitudinal measurements subject to outcome-dependent follow-up

A common problem for longitudinal data analyses is that subjects follow-up is irregular, often related to the past outcome or other factors associated with the outcome measure that are not included in the regression model. Analyses unadjusted for outcome-dependent follow-up yield biased estimates. We propose a longitudinal data analysis that can provide consistent estimates in regression models that are subject to outcome-dependent follow-up. We focus on semiparametric marginal log-link regression with arbitrary unspecified baseline function. Based on estimating equations, the proposed class of estimators are root n consistent and asymptotically normal. We present simulation studies that assess the performance of the estimators under finite samples. We illustrate our approach using data from a health services research study.     

Hypothesis testing in mixture regression models

Summary.  We establish asymptotic theory for both the maximum likelihood and the maximum modified likelihood estimators in mixture regression models. Moreover, under specific and reasonable conditions, we show that the optimal convergence rate of n ^−1/4 for estimating the mixing distribution is achievable for both the maximum likelihood and the maximum modified likelihood estimators. We also derive the asymptotic distributions of two log-likelihood ratio test statistics for testing homogeneity and we propose a resampling procedure for approximating the p -value. Simulation studies are conducted to investigate the empirical performance of the two test statistics. Finally, two real data sets are analysed to illustrate the application of our theoretical results.     

Performance of some new preliminary test ridge regression estimators and their properties

The problem of estimation of the regression coefficients in a multiple regression model is considered under multicollinearity situation when it is suspected that the regression coefficients may be restricted to a subspace. We present the estimators of the regression coefficients combining the idea of preliminary test and ridge regression methodology. Accordingly, we consider three estimators, namely, the unrestricted ridge regression estimator (URRE), the restricted ridge regression estimator (RRRE), and finally, the preliminary test ridge regression estimator (PTRRE). The biases, variancematrices and mean square errors (mse) of the estimators are derived and compared with the usual estimators. Regions of optimality of the estimators are determined by studying the mse criterion. The conditions of superiority of the estimators over the traditional estimators as in Saleh and Han (1990) and Ali and Saleh (1991) have also been discussed.     

Multiple-output quantile regression through optimal quantization

A new nonparametric quantile regression method based on the concept of optimal quantization was developed recently and was showed to provide estimators that often dominate their classical, kernel-type, competitors. In the present work, we extend this method to multiple-output regression problems. We show how quantization allows approximating population multiple-output regression quantiles based on halfspace depth. We prove that this approximation becomes arbitrarily accurate as the size of the quantization grid goes to infinity. We also derive a weak consistency result for a sample version of the proposed regression quantiles. Through simulations, we compare the performances of our estimators with (local constant and local bilinear) kernel competitors. The results reveal that the proposed quantization-based estimators, which are local constant in nature, outperform their kernel counterparts and even often dominate their local bilinear kernel competitors. The various approaches are also compared on artificial and real data.     

Estimation for semi-functional linear regression

This paper studies estimation in semi-functional linear regression. A general formulation is used to treat mean regression, median regression, quantile regression and robust mean regression in one setting. The linear slope function is estimated by the functional principal component basis and the nonparametric component is approximated by a B-spline function. The global convergence rates of the estimators of unknown slope function and nonparametric component are established under suitable norm. The convergence rate of the mean-squared prediction error for the proposed estimators is also established. Finite sample properties of our procedures are studied through Monte Carlo simulations. A real data example about Berkeley growth data is used to illustrate our proposed methodology.     

Nonstationary nonlinear quantile regression

This study examines estimation and inference based on quantile regression for parametric nonlinear models with an integrated time series covariate. We first derive the limiting distribution of the nonlinear quantile regression estimator and then consider testing for parameter restrictions, when the regression function is specified as an asymptotically homogeneous function. We also study linear-in-parameter regression models when the regression function is given by integrable regression functions as well as asymptotically homogeneous regression functions. We, furthermore, propose a fully modified estimator to reduce the bias in the original estimator under a certain set of conditions. Finally, simulation studies show that the estimators behave well, especially when the regression error term has a fat-tailed distribution.     

Randomized quantile regression estimation for heteroskedastic non parametric model

In this paper, we propose robust randomized quantile regression estimators for the mean and (condition) variance functions of the popular heteroskedastic non parametric regression model. Unlike classical approaches which consider quantile as a fixed quantity, our method treats quantile as a uniformly distributed random variable. Our proposed method can be employed to estimate the error distribution, which could significantly improve prediction results. An automatic bandwidth selection scheme will be discussed. Asymptotic properties and relative efficiencies of the proposed estimators are investigated. Our empirical results show that the proposed estimators work well even for random errors with infinite variances. Various numerical simulations and two real data examples are used to demonstrate our methodologies.     

A simulation study of five biased estimators for straight line regression

Five biased estimators of the slope in straight line regression are considered. For each, the estimate of the “bias parameter”, k, is a function of N, the number of observations, and [rcirc]² , the square of the least squares estimate of the standardized slope, β. The estimators include that of Farebrother, the ridge estimator of Hoerl, Kennard, and Baldwin, Vinod's shrunken estimators., and a new modification of one of the latter. Properties of the estimators are studied for 13 combinations of N and 3. Results of simulation experiments provide empirical evidence concerning the values of means and variances of the biased estimators of the slope and estimates of the “bias parameter”, the mean square errors of the estimators, and the frequency of improvement relative to least squares. Adjustments to degrees of freedom in the biased regression analysis of variance table are also considered. An extension of the new modification to the case of p> 1 independent variables is presented in an Appendix.