Two-Stage Welsh's Trimmed Mean for the Simultaneous Equations Model

This paper discusses the large sample theory of the two-stage Welsh's trimmed mean for the limited information simultaneous equations model. Besides having asymptotic normality, this trimmed mean, as the two-stage least squares estimator, is a generalized least squares estimator. It also acts as a robust Aitken estimator for the simultaneous equations model. Examples illustrate real data analysis and large sample inferences based on this trimmed mean.     

Robust ridge and robust Liu estimator for regression based on the LTS estimator

In the multiple linear regression analysis, the ridge regression estimator and the Liu estimator are often used to address multicollinearity. Besides multicollinearity, outliers are also a problem in the multiple linear regression analysis. We propose new biased estimators based on the least trimmed squares (LTS) ridge estimator and the LTS Liu estimator in the case of the presence of both outliers and multicollinearity. For this purpose, a simulation study is conducted in order to see the difference between the robust ridge estimator and the robust Liu estimator in terms of their effectiveness; the mean square error. In our simulations, the behavior of the new biased estimators is examined for types of outliers: X-space outlier, Y-space outlier, and X-and Y-space outlier. The results for a number of different illustrative cases are presented. This paper also provides the results for the robust ridge regression and robust Liu estimators based on a real-life data set combining the problem of multicollinearity and outliers.     

Trimmed least squares estimator as best trimmed linear conditional estimator for linear regression model

A class of trimmed linear conditional estimators based on regression quantiles for the linear regression model is introduced. This class serves as a robust analogue of non-robust linear unbiased estimators. Asymptotic analysis then shows that the trimmed least squares estimator based on regression quantiles ( Koenker and Bassett ( 1978 ) ) is the best in this estimator class in terms of asymptotic covariance matrices. The class of trimmed linear conditional estimators contains the Mallows-type bounded influence trimmed means ( see De Jongh et al ( 1988 ) ) and trimmed instrumental variables estimators. A large sample methodology based on trimmed instrumental variables estimator for confidence ellipsoids and hypothesis testing is also provided.     

Feasible robust estimator in restricted semiparametric regression models based on the LTS approach

Under some nonstochastic linear restrictions based on either additional information or prior knowledge in a semiparametric regression model, a family of feasible generalized robust estimators for the regression parameter is proposed. The least trimmed squares (LTS) method proposed by Rousseeuw as a highly robust regression estimator is a statistical technique for fitting a regression model based on the subset of h observations (out of n) whose least-square fit possesses the smallest sum of squared residuals. The coverage h may be set between n/2 and n. The LTS estimator involves computing the hyperplane that minimizes the sum of the smallest h squared residuals. For practical purpose, it is assumed that the covariance matrix of the error term is unknown and thus feasible estimators are replaced. Then, we develop an algorithm for the LTS estimator based on feasible methods. Through the Monte Carlo simulation studies and a real data example, performance of the feasible type of robust estimators is compared with the classical ones in restricted semiparametric regression models.     

Efficient Estimation and Robust Inference of Linear Regression Models in the Presence of Heteroscedastic Errors and High Leverage Points

It is common for linear regression models that the error variances are not the same for all observations and there are some high leverage data points. In such situations, the available literature advocates the use of heteroscedasticity consistent covariance matrix estimators (HCCME) for the testing of regression coefficients. Primarily, such estimators are based on the residuals derived from the ordinary least squares (OLS) estimator that itself can be seriously inefficient in the presence of heteroscedasticity. To get efficient estimation, many efficient estimators, namely the adaptive estimators are available but their performance has not been evaluated yet when the problem of heteroscedasticity is accompanied with the presence of high leverage data. In this article, the presence of high leverage data is taken into account to evaluate the performance of the adaptive estimator in terms of efficiency. Furthermore, our numerical work also evaluates the performance of the robust standard errors based on this efficient estimator in terms of interval estimation and null rejection rate (NRR).     

Robust two parameter ridge M-estimator for linear regression

The problem of multicollinearity and outliers in the data set produce undesirable effects on the ordinary least squares estimator. Therefore, robust two parameter ridge estimation based on M-estimator (ME) is introduced to deal with multicollinearity and outliers in the y-direction. The proposed estimator outperforms ME, two parameter ridge estimator and robust ridge M-estimator according to mean square error criterion. Moreover, a numerical example and a Monte Carlo simulation experiment are presented.     

A consistent simulation-based estimator in generalized linear mixed models

We propose a strongly root-n consistent simulation-based estimator for the generalized linear mixed models. This estimator is constructed based on the first two marginal moments of the response variables, and it allows the random effects to have any parametric distribution (not necessarily normal). Consistency and asymptotic normality for the proposed estimator are derived under fairly general regularity conditions. We also demonstrate that this estimator has a bounded influence function and that it is robust against data outliers. A bias correction technique is proposed to reduce the finite sample bias in the estimation of variance components. The methodology is illustrated through an application to the famed seizure count data and some simulation studies.     

A modified ridge m-estimator for linear regression model with multicollinearity and outliers

The ordinary least-square estimators for linear regression analysis with multicollinearity and outliers lead to unfavorable results. In this article, we propose a new robust modified ridge M-estimator (MRME) based on M-estimator (ME) to deal with the combined problem resulting from multicollinearity and outliers in the y-direction. MRME outperforms modified ridge estimator, robust ridge estimator and ME, according to mean squares error criterion. Furthermore, a numerical example and a Monte Carlo simulation experiment are given to illustrate some of the theoretical results.     

A Wald-type test statistic based on robust modified median estimator in logistic regression models

In this paper a new robust estimator, modified median estimator, is introduced and studied for the logistic regression model. This estimator is based on the median estimator considered in Hobza et al. [Robust median estimator in logistic regression. J Stat Plan Inference. 2008;138:3822–3840]. Its asymptotic distribution is obtained. Using the modified median estimator, we also consider a Wald-type test statistic for testing linear hypotheses in the logistic regression model and we obtain its asymptotic distribution under the assumption of random regressors. An extensive simulation study is presented in order to analyse the efficiency as well as the robustness of the modified median estimator and Wald-type test based on it.     

Robust estimation of the mean vector for high-dimensional data set using robust clustering

The first step in statistical analysis is the parameter estimation. In multivariate analysis, one of the parameters of interest to be estimated is the mean vector. In multivariate statistical analysis, it is usually assumed that the data come from a multivariate normal distribution. In this situation, the maximum likelihood estimator (MLE), that is, the sample mean vector, is the best estimator. However, when outliers exist in the data, the use of sample mean vector will result in poor estimation. So, other estimators which are robust to the existence of outliers should be used. The most popular robust multivariate estimator for estimating the mean vector is S-estimator with desirable properties. However, computing this estimator requires the use of a robust estimate of mean vector as a starting point. Usually minimum volume ellipsoid (MVE) is used as a starting point in computing S-estimator. For high-dimensional data computing, the MVE takes too much time. In some cases, this time is so large that the existing computers cannot perform the computation. In addition to the computation time, for high-dimensional data set the MVE method is not precise. In this paper, a robust starting point for S-estimator based on robust clustering is proposed which could be used for estimating the mean vector of the high-dimensional data. The performance of the proposed estimator in the presence of outliers is studied and the results indicate that the proposed estimator performs precisely and much better than some of the existing robust estimators for high-dimensional data.     

Risk performance of a pre-test estimator for normal variance with the Stein-variance estimator under the LINEX loss function

In this paper, we derive the exact formula of the risk function of a pre-test estimator for normal variance with the Stein-variance (PTSV) estimator when the asymmetric LINEX loss function is used. Fixing the critical value of the pre-test to unity which is a suggested critical value in some sense, we examine numerically the risk performance of the PTSV estimator based on the risk function derived. Our numerical results show that although the PTSV estimator does not dominate the usual variance estimator when under-estimation is more severe than over-estimation, the PTSV estimator dominates the usual variance estimator when over-estimation is more severe. It is also shown that the dominance of the PTSV estimator over the original Stein-variance estimator is robust to the extension from the quadratic loss function to the LINEX loss function.     

Robust estimation for the coefficient of a first order autoregressive process

Nonparametric methods, Theil's method and Hussain's method have been applied to simple linear regression problems for estimating the slope of the regression line.We extend these methods and propose a robust estimator to estimate the coefficient of a first order autoregressive process under various distribution shapes, A simulation study to compare Theil's estimator, Hus-sain's estimator, the least squares estimator, and the proposed estimator is also presented.     

Effects of model misspecification on tests of no randomized treatment effect arising from Cox's proportional hazards model

We examine the asymptotic and small sample properties of model-based and robust tests of the null hypothesis of no randomized treatment effect based on the partial likelihood arising from an arbitrarily misspecified Cox proportional hazards model. When the distribution of the censoring variable is either conditionally independent of the treatment group given covariates or conditionally independent of covariates given the treatment group, the numerators of the partial likelihood treatment score and Wald tests have asymptotic mean equal to 0 under the null hypothesis, regardless of whether or how the Cox model is misspecified. We show that the model-based variance estimators used in the calculation of the model-based tests are not, in general, consistent under model misspecification, yet using analytic considerations and simulations we show that their true sizes can be as close to the nominal value as tests calculated with robust variance estimators. As a special case, we show that the model-based log-rank test is asymptotically valid. When the Cox model is misspecified and the distribution of censoring depends on both treatment group and covariates, the asymptotic distributions of the resulting partial likelihood treatment score statistic and maximum partial likelihood estimator do not, in general, have a zero mean under the null hypothesis. Here neither the fully model-based tests, including the log-rank test, nor the robust tests will be asymptotically valid, and we show through simulations that the distortion to test size can be substantial.     

A two-step robust estimation of the process mean using M-estimator

Parameter estimation is the first step in constructing control charts. One of these parameters is the process mean. The classical estimators of the process mean are sensitive to the presence of outlying data and subgroups which contaminate the whole data. In existing robust estimators for the process mean, the effects of the presence of the individual outliers are being considered, while, in this paper, a robust estimator is being proposed to reduce the effect of outlying subgroups as well as the individual outliers within a subgroup. The proposed estimator was compared with some classical and robust estimators of the process mean. Although, its relative efficiency is fourth among the estimators tested, its robustness and efficiency are large when the outlying subgroups are present. Evaluation of the results indicated that the proposed estimator is less sensitive to the presence of outliers and the process mean performs well when there are no individual outliers or outlying subgroups.     

Inference for proportional hazard model with propensity score

Since the publication of the seminal paper by Cox (1972), proportional hazard model has become very popular in regression analysis for right censored data. In observational studies, treatment assignment may depend on observed covariates. If these confounding variables are not accounted for properly, the inference based on the Cox proportional hazard model may perform poorly. As shown in Rosenbaum and Rubin (1983), under the strongly ignorable treatment assignment assumption, conditioning on the propensity score yields valid causal effect estimates. Therefore we incorporate the propensity score into the Cox model for causal inference with survival data. We derive the asymptotic property of the maximum partial likelihood estimator when the model is correctly specified. Simulation results show that our method performs quite well for observational data. The approach is applied to a real dataset on the time of readmission of trauma patients. We also derive the asymptotic property of the maximum partial likelihood estimator with a robust variance estimator, when the model is incorrectly specified.     

Another look at Huber's estimator: A new minimax estimator in regression with stochastically bounded noise

Huber's estimator has had a long lasting impact, particularly on robust statistics. It is well known that under certain conditions, Huber's estimator is asymptotically minimax. A moderate generalization in rederiving Huber's estimator shows that Huber's estimator is not the only choice. We develop an alternative asymptotic minimax estimator and name it regression with stochastically bounded noise (RSBN). Simulations demonstrate that RSBN is slightly better in performance, although it is unclear how to justify such an improvement theoretically. We propose two numerical solutions: an iterative numerical solution, which is extremely easy to implement and is based on the proximal point method; and a solution by applying state-of-the-art nonlinear optimization software packages, e.g., SNOPT. Contribution: the generalization of the variational approach is interesting and should be useful in deriving other asymptotic minimax estimators in other problems.     

ROBUST ESTIMATION OF SMALL‐AREA MEANS AND QUANTILES

Small‐area estimation techniques have typically relied on plug‐in estimation based on models containing random area effects. More recently, regression M‐quantiles have been suggested for this purpose, thus avoiding conventional Gaussian assumptions, as well as problems associated with the specification of random effects. However, the plug‐in M‐quantile estimator for the small‐area mean can be shown to be the expected value of this mean with respect to a generally biased estimator of the small‐area cumulative distribution function of the characteristic of interest. To correct this problem, we propose a general framework for robust small‐area estimation, based on representing a small‐area estimator as a functional of a predictor of this small‐area cumulative distribution function. Key advantages of this framework are that it naturally leads to integrated estimation of small‐area means and quantiles and is not restricted to M‐quantile models. We also discuss mean squared error estimation for the resulting estimators, and demonstrate the advantages of our approach through model‐based and design‐based simulations, with the latter using economic data collected in an Australian farm survey.     

A Bayesian approach with generalized ridge estimation for high-dimensional regression and testing

This paper adopts a Bayesian strategy for generalized ridge estimation for high-dimensional regression. We also consider significance testing based on the proposed estimator, which is useful for selecting regressors. Both theoretical and simulation studies show that the proposed estimator can simultaneously outperform the ordinary ridge estimator and the LSE in terms of the mean square error (MSE) criterion. The simulation study also demonstrates the competitive MSE performance of our proposal with the Lasso under sparse models. We demonstrate the method using the lung cancer data involving high-dimensional microarrays.     

A revised Cholesky decomposition to combat multicollinearity in multiple regression models

As known, the ordinary least-squares estimator (OLSE) is unbiased and also, has the minimum variance among all the linear unbiased estimators. However, under multicollinearity the estimator is generally unstable and poor in the sense that variance of the regression coefficients may be inflated and absolute values of the estimates may be too large. There are several classes of biased estimators in statistical literature to decrease the effect of multicollinearity in the design matrix. Here, based on the Cholesky decomposition, we propose such an estimator which makes the data to be slightly distorted. The exact risk expressions as well as the biases are derived for the proposed estimator. Also, some results demonstrating superiority of the suggested estimator over OLSE are obtained. Finally, a Monté-Carlo simulation study and a real data application related to acetylene data are presented to support our theoretical discussions.     

Double robust estimator in general treatment regimes based on Covariate-balancing

Double robust estimators have double the chance of being a consistent estimator of a causal effect in binary treatments cases. In this paper, we proposed an estimator of a causal effect for general treatment regimes based on covariate-balancing. Under parametrical situation, our estimator has double robustness.