Estimation in multiple linear regression model with doubly censored data

In this article, we propose three M-estimators for multiple regression model when response variable is subject to double censoring. The consistency of the proposed M-estimators is established. A simulation study is conducted to investigate the performance of the proposed estimators. Furthermore, the simple bootstrap methods are used to construct interval estimators.     

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).     

Recursive M-estimators of location

This paper deals with recursive M-estimators of a location parameter θ in stationary processes when scale is regarded as a nuisance parameter. For the nonrecursive M-estimators, the median absolute deviation is a useful estimator of scale. Two recursive variants of the median absolute deviation are proposed and the performance of the resulting recursive estimators is examined in a numerical study.     

Tukey’s M-estimator of the Poisson parameter with a special focus on small means

We treat robust M-estimators for independent and identically distributed Poisson data. We introduce modified Tukey M-estimators with bias correction and compare them to M-estimators based on the Huber function as well as to weighted likelihood and other estimators by simulation in case of clean data and data with outliers. In particular, we investigate the problem of combining robustness and high efficiencies at small Poisson means caused by the strong asymmetry of such Poisson distributions and propose a further estimator based on adaptive trimming. The advantages of the constructed estimators are illustrated by an application to smoothing count data with a time varying mean and level shifts.     

Efficient M-estimators with auxiliary information

This paper introduces a new class of M-estimators based on generalised empirical likelihood (GEL) estimation with some auxiliary information available in the sample. The resulting class of estimators is efficient in the sense that it achieves the same asymptotic lower bound as that of the efficient generalised method of moment (GMM) estimator with the same auxiliary information. The paper also shows that in case of smooth estimating equations the proposed estimators enjoy a small second order bias property compared to both efficient GMM and full GEL estimators. Analytical formulae to obtain bias corrected estimators are also provided. Simulations show that with correctly specified auxiliary information the proposed estimators and in particular those based on empirical likelihood outperform standard M and efficient GMM estimators both in terms of finite sample bias and efficiency. On the other hand with moderately misspecified auxiliary information estimators based on the nonparametric tilting method are typically characterised by the best finite sample properties.     

ROBUST M-ESTIMATION OF THE INTRACLASS CORRELATION COEFFICIENT

Robust M-estimators of intraclass correlation coefficient, location and scale parameters are defined for familial data. It is shown that these estimators are strongly consistent. Also the asymptotic distributions of these estimators are derived when the underlying distribution is elliptically and permutationally symmetric.     

The combined dynamically weighted modified maximum likelihood estimators of the location and scale parameters

In this study, two new types of estimators of the location and scale parameters are proposed having high efficiency and robustness; the dynamically weighted modified maximum likelihood (DWMML) and the combined dynamically weighted modified maximum likelihood (CDWMML) estimators. Three pairs of the DWMML and two pairs of the CDWMML estimators of the location and scale parameters are produced, namely, the DWMML1, the DWMML2 and the DWMML3, and the CDWMML1 and the CDWMML2 estimators, respectively. Based on the simulation results, the DWMML1 estimators of the location and scale parameters are almost fully efficient (under normality) and robust at the same time. The DWMML3 estimators are asymptotically fully efficient and more robust than the M-estimators. The DWMML2 estimators are a compromise between efficiency and robustness. The CDWMML1 and CDWMML2 estimators are jointly very efficient and robust. Particularly, the CDWMML1 and CDWMML2 estimators of the scale parameter are superior compared to the other estimators of the scale parameter.     

M-estimators for isotonic regression

In this paper we propose a family of robust estimates for isotonic regression: isotonic M-estimators. We show that their asymptotic distribution is, up to an scalar factor, the same as that of Brunk's classical isotonic estimator. We also derive the influence function and the breakdown point of these estimates. Finally we perform a Monte Carlo study that shows that the proposed family includes estimators that are simultaneously highly efficient under Gaussian errors and highly robust when the error distribution has heavy tails.     

Some robust anova procedures under heteroscedasticity and nonnormality

In this paper we consider the problem of comparing several means under heteroscedasticity and nonnormality. By combining Huber‘s M-estimators with the Brown-Forsythe test, several robust procedures were developed; these procedures were compared through computer simulation studies with the Tan-Tabatabai procedure which was developed by combining Tiku's MML estimators with the Brown-Forsythe test. The numerical results indicate clearly that the Tan-Tabatabai procedure is considerably more powerful than tests based on Huber's M-estimators over a wide range of nonnormal distributions.     

Bounded-leverage weights for robust regression estimators

Both the least squares estimator and M-estimators of regression coefficients are susceptible to distortion when high leverage points occur among the predictor variables in a multiple linear regression model. In this article a weighting scheme which enables one to bound the leverage values of a weighted matrix of predictor variables is proposed. Bounded-leverage weighting of the predictor variables followed by M-estimation of the regression coefficients is shown to be effective in protecting against distortion due to extreme predictor-variable values, extreme response values, or outlier-induced multieollinearites. Bounded-leverage estimators can also protect against distortion by small groups of high leverage points.     

Diagnostic Checking for GARCH-Type Models

The asymptotic distributions of squared and absolute residual autocorrelations for GARCH model estimated by M-estimators are derived. Two diagnostic tests are developed which can be used to check the adequacy of GARCH model fitted by using M-estimators. Simulation results show that the empirical sizes of both tests are close to the nominal size in most of the cases. The power of test based on absolute residual autocorrelation is found better than test based on squared residual autocorrelations. Our results reveal that there are estimators that can fit GARCH-type models better than the commonly used quasi-maximum likelihood estimator under non normal errors. An application to real data set is also presented.     

M-estimators of some GARCH-type models; computation and application

In this paper, we consider robust M-estimation of time series models with both symmetric and asymmetric forms of heteroscedasticity related to the GARCH and GJR models. The class of estimators includes least absolute deviation (LAD), Huber’s, Cauchy and B-estimator as well as the well-known quasi maximum likelihood estimator (QMLE). Extensive simulations are used to check the relative performance of these estimators in both models and the weighted resampling methods are used to approximate the sampling distribution of M-estimators. Our study indicates that there are estimators that can perform better than QMLE and even outperform robust estimator such as LAD when the error distribution is heavy-tailed. These estimators are also applied to real data sets.     

A Robust and Almost Fully Efficient M-Estimator

Simultaneous robust estimates of location and scale parameters are derived from a class of M-estimating equations. A coefficient p ( p > 0), which plays a role similar to that of a tuning constant in the theory of M-estimation, determines the estimating equations. These estimating equations may be obtained as the gradient of a strictly convex criterion function. This article shows that the estimators are uniquely defined, asymptotically bi-variate normal and have positive breakdown for some choices of p . When p = 0.12 and p = 0.3, the estimators are almost fully efficient for normal and exponential distributions: efficiencies with respect to the maximum likelihood estimators are 1.00 and 0.99, respectively. It is shown that the location estimator for known scale has the maximum breakdown point 0.5 independent of p , when the target model is symmetric. Also it is shown that the scale estimator has a positive breakdown point which depends on the choice of p . A simulation study finds that the proposed location estimator has smaller variance than the Hodges–Lehmann estimator, Huber's minimax and bisquare M-estimators.     

A note on james type robust estimators

M-estimation of a single parameter of the life time distribution is considered based on independent and identically distributed survival data which may be randomly censored. The most robust and the optimal robust M-estimators of the location parameters of the survival time distribution are derived within a class considered in James (1986) as well as for the general unrestricted class. The properties of the estimators corresponding to the above two classes are discussed. A data set is used to illustrate the usefulness of the optimal robust estimators for the parameter of extreme value distribution.     

Robust Estimation for Parameters of the Extended Burr Type III Distribution

We consider various robust estimators for the extended Burr Type III (EBIII) distribution for complete data with outliers. The considered robust estimators are M-estimators, least absolute deviations, Theil, Siegel's repeated median, least trimmed squares, and least median of squares. Before we perform the aforementioned estimators for the EBIII, we adapt the quantiles method to the estimation of the shape parameter k of the EBIII. The simulation results show that the considered robust estimators generally outperform the existing estimation approaches for data with upper outliers, with certain of them retaining a relatively high degree of efficiency for small sample sizes.     

Fast and robust estimators of variance components in the nested error model

Usual fitting methods for the nested error linear regression model are known to be very sensitive to the effect of even a single outlier. Robust approaches for the unbalanced nested error model with proved robustness and efficiency properties, such as M-estimators, are typically obtained through iterative algorithms. These algorithms are often computationally intensive and require robust estimates of the same parameters to start the algorithms, but so far no robust starting values have been proposed for this model. This paper proposes computationally fast robust estimators for the variance components under an unbalanced nested error model, based on a simple robustification of the fitting-of-constants method or Henderson method III. These estimators can be used as starting values for other iterative methods. Our simulations show that they are highly robust to various types of contamination of different magnitude.     

ON THE ROBUST ANALYSIS OF VARIANCE COMPONENTS MODELS FOR PEDIGREE DATA

Quantitative traits measured over pedigrees of individuals may be analysed using maximum likelihood estimation, assuming that the trait has a multivariate normal distribution. This approach is often used in the analysis of mixed linear models. In this paper a robust version of the log likelihood for multivariate normal data is used to construct M-estimators which are resistant to contamination by outliers. The robust estimators are found using a minimisation routine which retains the flexible parameterisations of the multivariate normal approach. Asymptotic properties of the estimators are derived, computation of the estimates and their use in outlier detection tests are discussed, and a small simulation study is conducted.     

Asymptotic behavior of iterative m-estimators for the linear model

In this paper, we study the M-estimators for the linear model when they are computed by a class of numerical iterative procedures. This class includes the usual method of Newton-Raphson, iteratively reweighted least squares and iterative winsorization. We show that under mild conditions, the numerical iterative procedures converge and the resulting estimators are consistent and asymptotically normal.     

ROBUST RIDGE REGRESSION BASED ON AN M-ESTIMATOR

Consider the linear regression model y =β₀1 +Xβ+ in the usual notation. It is argued that the class of ordinary ridge estimators obtained by shrinking the least squares estimator by the matrix (X¹X + kI)^-1X'X is sensitive to outliers in the ^variable. To overcome this problem, we propose a new class of ridge-type M-estimators, obtained by shrinking an M-estimator (instead of the least squares estimator) by the same matrix. Since the optimal value of the ridge parameter k is unknown, we suggest a procedure for choosing it adaptively. In a reasonably large scale simulation study with a particular M-estimator, we found that if the conditions are such that the M-estimator is more efficient than the least squares estimator then the corresponding ridge-type M-estimator proposed here is better, in terms of a Mean Squared Error criteria, than the ordinary ridge estimator with k chosen suitably. An example illustrates that the estimators proposed here are less sensitive to outliers in the y-variable than ordinary ridge estimators.     

ROBUST ESTIMATION UNDER TYPE II CENSORING

The properties of robust M-estimators with type II censored failure time data are considered. The optimal members within two classes of ψ-functions are characterized. The first optimality result is the censored data analogue of the optimality result described in Hampel et al. (1986); the estimators corresponding to the optimal members within this class are referred to as the optimal robust estimators. The second result pertains to a restricted class of ψ-functions which is the analogue of the class of ψ-functions considered in James (1986) for randomly censored data; the estimators corresponding to the optimal members within this restricted class are referred to as the optimal James-type estimators. We examine the usefulness of the two classes of ψ-functions and find that the breakdown point and efficiency of the optimal James-type estimators compare favourably with those of the corresponding optimal robust estimators. From the computational point of view, the optimal James-type ψ-functions are readily obtainable from the optimal ψ-functions in the uncensored case. The ψ-functions for the optimal robust estimators require a separate algorithm which is provided. A data set illustrates the optimal robust estimators for the parameters of the extreme value distribution.