Sparse least trimmed squares regression.

Sparse model estimation is a topic of high importance in modern data analysis due to the increasing availability of data sets with a large number of variables. Another common problem in applied statistics is the presence of outliers in the data. This paper combines robust regression and sparse model estimation. A robust and sparse estimator is introduced by adding an L1 penalty on the coefficient estimates to the well known least trimmed squares (LTS) estimator. The breakdown point of this sparse LTS estimator is derived, and a fast algorithm for its computation is proposed. Both the simulation study and the real data example show that the LTS has better prediction performance than its competitors in the presence of leverage points.Breakdown point; Outliers; Penalized regression; Robust regression; Trimming;

The multivariate least trimmed squares estimator.

In this paper we introduce the least trimmed squares estimator for multivariate regression. We give three equivalent formulations of the estimator and obtain its breakdown point. A fast algorithm for its computation is proposed. We prove Fisher-consistency at the multivariate regression model with elliptically symmetric error distribution and derive the influence function. Simulations investigate the finite-sample efficiency and robustness of the estimator. To increase the efficiency of the estimator, we also consider a one-step reweighted version, as well as multivariate generalizations of one-step GM-estimators.Model; Data; Distribution; Simulation;

Robust artificial neural networks and outlier detection. Technical report

Large outliers break down linear and nonlinear regression models. Robust regression methods allow one to filter out the outliers when building a model. By replacing the traditional least squares criterion with the least trimmed squares criterion, in which half of data is treated as potential outliers, one can fit accurate regression models to strongly contaminated data. High-breakdown methods have become very well established in linear regression, but have started being applied for non-linear regression only recently. In this work, we examine the problem of fitting artificial neural networks to contaminated data using least trimmed squares criterion. We introduce a penalized least trimmed squares criterion which prevents unnecessary removal of valid data. Training of ANNs leads to a challenging non-smooth global optimization problem. We compare the efficiency of several derivative-free optimization methods in solving it, and show that our approach identifies the outliers correctly when ANNs are used for nonlinear regression

Asymptotics of Least Trimmed Squares Regression

High breakdown-point regression estimators protect against large errors both in explanatory and dependent variables.The least trimmed squares (LTS) estimator is one of frequently used, easily understandable, and thoroughly studied (from the robustness point of view) high breakdown-point estimators.In spite of its increasing popularity and number of applications, there are only conjectures and hints about its asymptotic behavior in regression after two decades of its existence.We derive here all important asymptotic properties of LTS, including the asymptotic normality and variance, under mild B-mixing conditions.

Asymptotics of Least Trimmed Squares Regression

High breakdown-point regression estimators protect against large errors both in explanatory and dependent variables.The least trimmed squares (LTS) estimator is one of frequently used, easily understandable, and thoroughly studied (from the robustness point of view) high breakdown-point estimators.In spite of its increasing popularity and number of applications, there are only conjectures and hints about its asymptotic behavior in regression after two decades of its existence.We derive here all important asymptotic properties of LTS, including the asymptotic normality and variance, under mild B-mixing conditions.least squares;estimation;regression analysis

Letter to the Editor

The paper by Alfons, Croux and Gelper (2013), Sparse least trimmed squares regression for analyzing high-dimensional large data sets, considered a combination of least trimmed squares (LTS) and lasso penalty for robust and sparse high-dimensional regression. In a recent paper [She and Owen (2011)], a method for outlier detection based on a sparsity penalty on the mean shift parameter was proposed (designated by "SO" in the following). This work is mentioned in Alfons et al. as being an "entirely different approach." Certainly the problem studied by Alfons et al. is novel and interesting.Comment: Published in at http://dx.doi.org/10.1214/13-AOAS640 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Asymptotics for the least trimmed squares estimator

Novel properties of the objective function in both empirical and population settings of the least trimmed squares (LTS) regression (Rousseeuw 1984), along with other properties of the LTS, are established first time in this article. The primary properties of the objective function facilitate the establishment of other original results, including influence function and Fisher consistency. The strong consistency is established first time with the help of a generalized Glivenko-Cantelli Theorem over a class of functions. Differentiability and stochastic equicontinuity promote the re-establishment of asymptotic normality with a neat, concise, and novel approach.Comment: 20 pages. arXiv admin note: substantial text overlap with arXiv:2203.10387, arXiv:2204.0070

Reweighted Least Trimmed Squares: An Alternative to One-Step Estimators

A new class of robust regression estimators is proposed that forms an alternative to traditional robust one-step estimators and that achieves the √n rate of convergence irrespective of the initial estimator under a wide range of distributional assumptions. The proposed reweighted least trimmed squares (RLTS) estimator employs data-dependent weights determined from an initial robust fit. Just like many existing one- and two-step robust methods, the RLTS estimator preserves robust properties of the initial robust estimate. However contrary to existing methods, the first-order asymptotic behavior of RLTS is independent of the initial estimate even if errors exhibit heteroscedasticity, asymmetry, or serial correlation. Moreover, we derive the asymptotic distribution of RLTS and show that it is asymptotically efficient for normally distributed errors. A simulation study documents benefits of these theoretical properties in finite samples.asymptotic efficiency;breakdown point;least trimmed squares

Trust and Growth in the 1990s: A Robustness Analysis

We conduct an extensive robustness analysis of the relationship between trust and growth for a later time period (the 1990s) and with a bigger sample (63 countries) than previous studies. In addition to robustness tests that focus on model uncertainty, we use Least Trimmed Squares, a robust estimation technique, to identify outliers and investigate how they affect the results. We find that the trust-growth relationship is less robust with respect to empirical specification and to countries in the sample than previously claimed, and that outliers affect the results. Nevertheless trust seems quite important compared with many other growth-regression variables.trust; growth; robustness analysis; extreme bounds analysis; social capital; least trimmed squares; outliers