On the use of robust regression in econometrics

The use of robust regression estimators has gained popularity among applied econometricians. The main argument invoked to justify the use of the robust estimators is that they provide efficiency gains in the presence of outliers or non-normal errors. Unfortunately, most practitioners seem to be unaware of the fact that heteroskedastic and skewed errors can dramatically affect the properties of these estimators. In this paper we reconsider the interpretation of the specific robust estimator that has become popular in applied econometrics, and conclude that its use in this context cannot be generally recommended.

Robustness analysis of a Maximum Correntropy framework for linear regression

In this paper we formulate a solution of the robust linear regression problem in a general framework of correntropy maximization. Our formulation yields a unified class of estimators which includes the Gaussian and Laplacian kernel-based correntropy estimators as special cases. An analysis of the robustness properties is then provided. The analysis includes a quantitative characterization of the informativity degree of the regression which is appropriate for studying the stability of the estimator. Using this tool, a sufficient condition is expressed under which the parametric estimation error is shown to be bounded. Explicit expression of the bound is given and discussion on its numerical computation is supplied. For illustration purpose, two special cases are numerically studied.Comment: 10 pages, 5 figures, To appear in Automatic

Asymptotic theory for iterated one-step Huber-skip estimators

Iterated one-step Huber-skip M-estimators are considered for regression problems. Each one-step estimator is a reweighted least squares estimators with zero/one weights determined by the initial estimator and the data. The asymptotic theory is given for iteration of such estimators using a tightness argument. The results apply to stationary as well as non-stationary regression problems.Huber-skip; iteration; one-step M-estimators; unit roots

Robust regularized singular value decomposition with application to mortality data

We develop a robust regularized singular value decomposition (RobRSVD) method for analyzing two-way functional data. The research is motivated by the application of modeling human mortality as a smooth two-way function of age group and year. The RobRSVD is formulated as a penalized loss minimization problem where a robust loss function is used to measure the reconstruction error of a low-rank matrix approximation of the data, and an appropriately defined two-way roughness penalty function is used to ensure smoothness along each of the two functional domains. By viewing the minimization problem as two conditional regularized robust regressions, we develop a fast iterative reweighted least squares algorithm to implement the method. Our implementation naturally incorporates missing values. Furthermore, our formulation allows rigorous derivation of leave-one-row/column-out cross-validation and generalized cross-validation criteria, which enable computationally efficient data-driven penalty parameter selection. The advantages of the new robust method over nonrobust ones are shown via extensive simulation studies and the mortality rate application.Comment: Published in at http://dx.doi.org/10.1214/13-AOAS649 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Discriminative Density-ratio Estimation

The covariate shift is a challenging problem in supervised learning that results from the discrepancy between the training and test distributions. An effective approach which recently drew a considerable attention in the research community is to reweight the training samples to minimize that discrepancy. In specific, many methods are based on developing Density-ratio (DR) estimation techniques that apply to both regression and classification problems. Although these methods work well for regression problems, their performance on classification problems is not satisfactory. This is due to a key observation that these methods focus on matching the sample marginal distributions without paying attention to preserving the separation between classes in the reweighted space. In this paper, we propose a novel method for Discriminative Density-ratio (DDR) estimation that addresses the aforementioned problem and aims at estimating the density-ratio of joint distributions in a class-wise manner. The proposed algorithm is an iterative procedure that alternates between estimating the class information for the test data and estimating new density ratio for each class. To incorporate the estimated class information of the test data, a soft matching technique is proposed. In addition, we employ an effective criterion which adopts mutual information as an indicator to stop the iterative procedure while resulting in a decision boundary that lies in a sparse region. Experiments on synthetic and benchmark datasets demonstrate the superiority of the proposed method in terms of both accuracy and robustness

Robust PCA as Bilinear Decomposition with Outlier-Sparsity Regularization

Principal component analysis (PCA) is widely used for dimensionality reduction, with well-documented merits in various applications involving high-dimensional data, including computer vision, preference measurement, and bioinformatics. In this context, the fresh look advocated here permeates benefits from variable selection and compressive sampling, to robustify PCA against outliers. A least-trimmed squares estimator of a low-rank bilinear factor analysis model is shown closely related to that obtained from an $\ell_0$-(pseudo)norm-regularized criterion encouraging sparsity in a matrix explicitly modeling the outliers. This connection suggests robust PCA schemes based on convex relaxation, which lead naturally to a family of robust estimators encompassing Huber's optimal M-class as a special case. Outliers are identified by tuning a regularization parameter, which amounts to controlling sparsity of the outlier matrix along the whole robustification path of (group) least-absolute shrinkage and selection operator (Lasso) solutions. Beyond its neat ties to robust statistics, the developed outlier-aware PCA framework is versatile to accommodate novel and scalable algorithms to: i) track the low-rank signal subspace robustly, as new data are acquired in real time; and ii) determine principal components robustly in (possibly) infinite-dimensional feature spaces. Synthetic and real data tests corroborate the effectiveness of the proposed robust PCA schemes, when used to identify aberrant responses in personality assessment surveys, as well as unveil communities in social networks, and intruders from video surveillance data.Comment: 30 pages, submitted to IEEE Transactions on Signal Processin