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Consistency and robustness of kernel-based regression in convex risk minimization

Andreas Christmann, Ingo Steinwart
2007 Bernoulli  
Then we consider robustness properties of such kernel methods.  ...  We investigate statistical properties for a broad class of modern kernel-based regression (KBR) methods.  ...  Interestingly, similar tail properties of Y are widely used to obtain consistency of nonparametric regression estimators and to establish robustness properties of M estimators in linear regression.  ... 
doi:10.3150/07-bej5102 fatcat:spm5em6jv5hpdp4uokz6ssajee

Adaptive Hard Thresholding for Near-optimal Consistent Robust Regression [article]

Arun Sai Suggala, Kush Bhatia, Pradeep Ravikumar, Prateek Jain
2019 arXiv   pre-print
We study the problem of robust linear regression with response variable corruptions.  ...  We also extend our estimator to robust sparse linear regression and show that similar guarantees hold in this setting.  ...  Main Results Robust Regression: For robust regression with oblivious response variable corruptions, we propose the first efficient consistent estimator with break-down point of 1.  ... 
arXiv:1903.08192v1 fatcat:pyqsvm467rg5vcilbjbt5kdw6q

Robustness Versus Consistency in Ill-Posed Classification and Regression Problems [chapter]

Robert Hable, Andreas Christmann
2012 Studies in Classification, Data Analysis, and Knowledge Organization  
weaker properties: consistency ; risk-consistency robustness ; risk-robustness Regression/Classification: (X 1 , Y 1 ), . . . , (X n , Y n ) ∼ P 0 i.i.d.  ...  weaker properties: consistency ; risk-consistency robustness ; risk-robustness Regression/Classification: (X 1 , Y 1 ), . . . , (X n , Y n ) ∼ P 0 i.i.d.  ...  Support vector machine (SVM) SVMs can be (depending on λn ∈ (0, ∞), n ∈ N) either (risk-)consistent or qualitatively (risk-)robust  ... 
doi:10.1007/978-3-642-28894-4_4 fatcat:zq7xxk4ztfdydneeivzvyb7ace

Robust Heteroscedasticity Consistent Covariance Matrix Estimator based on Robust Mahalanobis Distance and Diagnostic Robust Generalized Potential Weighting Methods in Linear Regression

M. Habshah, Muhammad Sani, Jayanthi Arasan
2018 Journal of Modern Applied Statistical Methods  
based on index set equality (DRGP(ISE)) on robust heteroscedasticity consistent covariance matrix estimators.  ...  Based on Furno (1996) , two robust weighting methods are proposed based on HLP detection measures (robust Mahalanobis distance based on minimum volume ellipsoid and diagnostic robust generalized potential  ...  Furno (1996) proposed the robust heteroscedasticity consistent covariance matrix (RHCCM) in order to reduce the biased caused by leverage points.  ... 
doi:10.22237/jmasm/1530279855 fatcat:q35fvrlhh5evlf4yy3hvtv3dcy

Robust Regression Shrinkage and Consistent Variable Selection Through the LAD-Lasso

Hansheng Wang, Guodong Li, Guohua Jiang
2007 Journal of business & economic statistics  
The least absolute deviation (LAD) regression is a useful method for robust regression, and the least absolute shrinkage and selection operator (lasso) is a popular choice for shrinkage estimation and  ...  Compared with the LAD regression, LAD-lasso can do parameter estimation and variable selection simultaneously.  ...  In this article we attempt to develop a robust regression shrinkage and selection method that can do regression shrinkage and selection (like lasso) and is also resistant to outliers or heavy-tailed errors  ... 
doi:10.1198/073500106000000251 fatcat:po5dng6h5fbkfhz4ltmbqvtoxq

I Didn't Run a Single Regression

Christian Müller
2006 Social Science Research Network  
Recent research therefore aims at obtaining robust regression results by systematically running multiple models and picking surviving variables.  ...  This note shows that a very popular of these approaches, the robust regression due to Sala-i-Martin (1997) very likely leads to inconsistent conclusions but may be remedied by refining the 'testimation  ...  We may now calculate the share of admissible, that is consistent, regressions of the total number of regressions.  ... 
doi:10.2139/ssrn.881717 fatcat:3pq6dskrkjho3chbeynerishhi

Variance estimation for logistic regression in case-cohort studies [article]

Hisashi Noma
2022 arXiv   pre-print
The logistic regression analysis proposed by Schouten et al.  ...  Through simulation studies, the bootstrap method consistently provided more precise confidence intervals compared with those provided by the robust variance method, while retaining adequate coverage probabilities  ...  consistently estimated by the robust variance estimator.  ... 
arXiv:2208.03474v2 fatcat:ly3mqvmqqzf6tmkfhx2r5tnpcq

Cellwise Robust M Regression [article]

Peter Filzmoser, Sebastiaan Höppner, Irene Ortner, Sven Serneels and Tim Verdonck
2020 arXiv   pre-print
The cellwise robust M regression estimator is introduced as the first estimator of its kind that intrinsically yields both a map of cellwise outliers consistent with the linear model, and a vector of regression  ...  The method is illustrated to be equally robust as its casewise counterpart, MM regression. The cellwise regression method discards less information than any casewise robust estimator.  ...  MM regression estimators consist of two steps: at first, a highly robust initial estimate is calculated, which conveys its high breakdown point to the entire procedure.  ... 
arXiv:1912.03407v2 fatcat:coena4r4x5cljijj3ogpqug7hm

Standard errors estimation in the presence of high leverage point and heteroscedastic errors in multiple linear regression

Khoo Li Peng, Robiah Adnan, Maizah Hura Ahmad
2014 Malaysian Journal of Fundamental and Applied Sciences  
Robust Heteroscedastic Consistent Covariance Matrix (RHCCM) is the combination of a robust method and Heteroscedasticit Consistent Covariance Matrix (HCCM).  ...  In this study, the Robust Heteroscedastic Consistent Covariance Matrix (RHCCM) was proposed in order to estimate standard errors of regression coefficients in the presence of high leverage points and heteroscedastic  ...  Robust heteroscedasticity consistent covariance matrix Heteroscedasticity Consistent Covariance Matrix (HCCM) estimators are derived from an estimate of variance-covariance matrix of the regression coefficient  ... 
doi:10.11113/mjfas.v10n3.279 fatcat:wmrvjgvjtrdk7idwfolc5ewmdu

A Polynomial-time Form of Robust Regression

Yaoliang Yu, Özlem Aslan, Dale Schuurmans
2012 Neural Information Processing Systems  
We develop an estimator that requires only polynomial-time, while achieving certain robustness and consistency guarantees.  ...  We present a general formulation for robust regression-Variational M-estimation-that unifies a number of robust regression methods while allowing a tractable approximation strategy.  ...  Thus the need for regression estimators that are both scalable and robust is increasing.  ... 
dblp:conf/nips/YuAS12 fatcat:6cgrlwtgjnf6ta3cl7nyc6gtn4

Interpolation can hurt robust generalization even when there is no noise [article]

Konstantin Donhauser, Alexandru Ţifrea, Michael Aerni, Reinhard Heckel, Fanny Yang
2021 arXiv   pre-print
We prove this phenomenon for the robust risk of both linear regression and classification and hence provide the first theoretical result on robust overfitting.  ...  Min-2 -norm interpolation in robust linear regression In the context of regression, we illustrate overfitting of the robust risk in Equation ( 2 ) with the set of consistent 2 -perturbations U 2 ( ).  ...  Robust overfitting in noiseless linear regression The following theorem provides a precise asymptotic expression of the robust risk under consistent Theorem 3.1.  ... 
arXiv:2108.02883v2 fatcat:xeamcarrwfed3pfh2enaauys4i

Robust Function-on-Function Regression

Harjit Hullait, David S. Leslie, Nicos G. Pavlidis, Steve King
We therefore introduce a Fisher-consistent robust functional linear regression model that is able to effectively fit data in the presence of outliers.  ...  We therefore introduce a consistent robust model selection procedure to choose the number of principal components.  ...  Assume C1-C6 hold then the robust regression functionβ(s, t) is Fisher- consistent.  ... 
doi:10.6084/m9.figshare.12844308.v1 fatcat:3walhc4warbfposyjfjced3jz4

A Unified Robust Regression Model for Lasso-like Algorithms

Wenzhuo Yang, Huan Xu
2013 International Conference on Machine Learning  
Using this robustness interpretation, we present new sparsity results, and establish the statistical consistency of the proposed regularized linear regression.  ...  We develop a unified robust linear regression model and show that it is equivalent to a general regularization framework to encourage sparse-like structure that contains group Lasso and fused Lasso as  ...  In particular, they showed that Lasso is equivalent to a robust linear regression formulation, and such robustness interpretation implies the sparsity and the consistency of Lasso.  ... 
dblp:conf/icml/YangX13 fatcat:bftcs3t6mndblo2rawqcpjprqq

Robust Three-Step Regression Based on Comedian and Its Performance in Cell-Wise and Case-Wise Outliers

Henry Velasco, Henry Laniado, Mauricio Toro, Víctor Leiva, Yuhlong Lio
2020 Mathematics  
In this work, a robust estimator is proposed based on a three-step method named 3S-regression, which uses the comedian as a highly robust scatter estimate.  ...  Few methods have been developed in order to deal with both types of outliers when formulating a regression model.  ...  The Consistent Factor in MAD The MAD is a very robust scatter estimate, which has 50% breakdown point (the best possible).  ... 
doi:10.3390/math8081259 fatcat:wu6u7m2isbeldak3bzey4rnp4u

Robust estimation in the errors-in-variables model

1989 Biometrika  
These estimates are shown to be consistent at elliptical errors-in-variables models and robust, if the corresponding loss function is bounded.  ...  Some key words: Errors-in-variables; M estimate; Orthogonal regression; Robustness. 1.  ...  in this case consistency and robustness can be preserved.  ... 
doi:10.1093/biomet/76.1.149 fatcat:qchxywbvwbcv5b2qbdi7gyvrgy
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