High finite-sample efficiency and robustness based on distance-constrained maximum likelihood

Good robust estimators can be tuned to combine a high breakdown point and a specified asymptotic efficiency at a central model. This happens in regression with MM- and -estimators among others. However, the finite-sample efficiency of these estimators can be much lower than the asymptotic one. To overcome this drawback, an approach is proposed for parametric models, which is based on a distance between parameters. Given a robust estimator, the proposed one is obtained by maximizing the likelihood under the constraint that the distance is less than a given threshold. For the linear model with normal errors, simulations show that the proposed estimator attains a finite-sample efficiency close to one while improving the robustness of the initial estimator. The same approach also shows good results in the estimation of multivariate location and scatter.Facultad de Ciencias Exacta

Forecasting Multiple Time Series With One-Sided Dynamic Principal Components

We define one-sided dynamic principal components (ODPC) for time series as linear combinations of the present and past values of the series that minimize the reconstruction mean squared error. Usually dynamic principal components have been defined as functions of past and future values of the series and therefore they are not appropriate for forecasting purposes. On the contrary, it is shown that the ODPC introduced in this article can be successfully used for forecasting high-dimensional multiple time series. An alternating least-squares algorithm to compute the proposed ODPC is presented. We prove that for stationary and ergodic time series the estimated values converge to their population analogs. We also prove that asymptotically, when both the number of series and the sample size go to infinity, if the data follow a dynamic factor model, the reconstruction obtained with ODPC converges in mean square to the common part of the factor model. The results of a simulation study show that the forecasts obtained with ODPC compare favorably with those obtained using other forecasting methods based on dynamic factor models.Fil: Peña, Daniel. Universidad Carlos III de Madrid. Instituto de Salud; EspañaFil: Smucler, Ezequiel. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Cálculo; ArgentinaFil: Yohai, Victor Jaime. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Cálculo; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentin

Highly Robust and Highly Finite Sample Efficient Estimators for the Linear Model

In this paper, we propose a new family of robust regression estimators, which we call bounded residual scale estimators (BRS-estimators) which are simultaneously highly robust and highly efficient for small samples with normally distributed errors. To define these estimators it is required to have a robust M-scale and a family of robust MM-estimators. We start by choosing in this family a highly robust initial estimator but not necessarily highly efficient. Loosely speaking, the BRS-estimator is defined as the estimator in the MM family which is closest to the LSE among those with a robust M-scale sufficiently close to the one of the initial estimators. The efficiency of the BRS is derived from the fact that when there are not outliers in the sample and the errors are normally distributed, the scale of the LSE is similar to the one of the initial estimator. The robustness of the BRS-estimator comes from the fact that its robust scale is close to the one of the initial highly robust estimator. The results of a Monte Carlo study show that the proposed estimator has a high finite-sample efficiency, and is highly resistant to outlier contamination.Fil: Smucler, Ezequiel. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Calculo. - Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Calculo; ArgentinaFil: Yohai, Victor Jaime. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Calculo. - Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Calculo; Argentin

Robust and sparse estimators for linear regression models

Penalized regression estimators are popular tools for the analysis of sparse and high-dimensional models. However, penalized regression estimators defined using an unbounded loss function can be very sensitive to the presence of outlying observations, especially to high leverage outliers. The robust and asymptotic properties of ℓ1-penalized MM-estimators and MM-estimators with an adaptive ℓ1 penalty are studied. For the case of a fixed number of covariates, the asymptotic distribution of the estimators is derived and it is proven that for the case of an adaptive ℓ1 penalty, the resulting estimator can have the oracle property. The advantages of the proposed estimators are demonstrated through an extensive simulation study and the analysis of real data sets. The proofs of the theoretical results are available in the Supplementary material to this article (see Appendix A).Fil: Smucler, Ezequiel. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Cálculo; ArgentinaFil: Yohai, Victor Jaime. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Cálculo; Argentin

Generalized Dynamic Principal Components

Brillinger defined dynamic principal components (DPC) for time series based on a reconstruction criterion. He gave a very elegant theoretical solution and proposed an estimator which is consistent under stationarity. Here, we propose a new enterally empirical approach to DPC. The main differences with the existing methods—mainly Brillinger procedure—are (1) the DPC we propose need not be a linear combination of the observations and (2) it can be based on a variety of loss functions including robust ones. Unlike Brillinger, we do not establish any consistency results; however, contrary to Brillinger’s, which has a very strong stationarity flavor, our concept aims at a better adaptation to possible nonstationary features of the series. We also present a robust version of our procedure that allows to estimate the DPC when the series have outlier contamination. We give iterative algorithms to compute the proposed procedures that can be used with a large number of variables. Our nonrobust and robust procedures are illustrated with real datasets. Supplementary materials for this article are available online.Fil: Peña, Daniel. Universidad Carlos III de Madrid. Instituto de Salud; EspañaFil: Yohai, Victor Jaime. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Robust estimation for vector autoregressive models

A new class of robust estimators for VAR models is introduced. These estimators are an extension to the multivariate case of the MM-estimators based on a bounded innovation propagation AR model. They have a filtering mechanism that avoids the propagation of the effect of one outlier to the residuals of the subsequent periods. Besides, they are consistent and have the same asymptotic normal distribution as regular MM-estimators for VAR models. A Monte Carlo study shows that these estimators compare favorable with respect to other robust ones.Fil: Muler, Nora. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; ArgentinaFil: Yohai, Victor Jaime. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Robust functional linear regression based on splines

Many existing methods for functional regression are based on the minimization of an L2 norm of the residuals and are therefore sensitive to atypical observations, which may affect the predictive power and/or the smoothness of the resulting estimate. A robust version of a spline-based estimate is presented, which has the form of an MM estimate, where the L2 loss is replaced by a bounded loss function. The estimate can be computed by a fast iterative algorithm. The proposed approach is compared, with favorable results, to the one based on L2 and to both classical and robust Partial Least Squares through an example with high-dimensional real data and a simulation study.Fil: Maronna, Ricardo A.. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matematicas; ArgentinaFil: Yohai, Victor Jaime. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

M estimators based on the probability integral transformation with applications to count data

M estimators based on the probability integral transformation for discrete distributions are introduced and their asymptotic properties are proved. The proposed estimators are applied to count data in a simulation study and in a real data set of hospital lengths of stay.Fil: Valdora, Marina Silvia. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Calculo. - Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Calculo; ArgentinaFil: Yohai, Victor Jaime. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Calculo. - Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Calculo; Argentin

Correcting MM estimates for "fat" data sets

Regression MM estimates require the estimation of the error scale, and the determination of a constant that controls the efficiency. These two steps are based on the asymptotic results that are derived assuming that the number of predictors p remains fixed while the number of observations n tends to infinity, which means assuming that the ratio p/n is "small". However, many high-dimensional data sets have a "large" value of p/n (say, ≥0.2). It is shown that the standard asymptotic results do not hold if p/n is large; namely that (a) the estimated scale underestimates the true error scale, and (b) that even if the scale is correctly estimated, the actual efficiency can be much lower than the nominal one. To overcome these drawbacks simple corrections for the scale and for the efficiency controlling constant are proposed, and it is demonstrated that these corrections improve on the estimate's performance under both normal and contaminated data.Fil: Maronna, Ricardo Antonio. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas; ArgentinaFil: Yohai, Victor Jaime. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Robust location estimation with missing data

In a missing data setting, we have a sample in which a vector of explanatory variables xi is observed for every subject i, while scalar responses yi are missing by happenstance on some individuals. In this work we propose robust estimators of the distribution of the responses assuming missing at random (MAR) data, under a semiparametric regression model. Our approach allows the consistent estimation of any weakly continuous functional of the response’s distribution. In particular, strongly consistent estimators of any continuous location functional, such as the median, L-functionals and M-functionals, are proposed. A robust fit for the regression model combined with the robust properties of the location functional gives rise to a robust recipe for estimating the location parameter. Robustness is quantified through the breakdown point of the proposed procedure. The asymptotic distribution of the location estimators is also derived. The proofs of the theorems are presented in Supplementary Material available online.Avec les donnees manquantes, nous avons un ´ echantillon pour lequel les variables explicatives ´ xi sont observees pour chaque sujet ´ i, tandis que les variables reponses ´ yi sont manquantes au hasard pour quelques individus. Dans ce travail, nous proposons des estimateurs robustes pour la fonction de distribution des variables reponses en supposant que les donn ´ ees soient manquantes au hasard (MAR), sous un mod ´ ele ` de regression non param ´ etrique. Notre approche permet l’estimation coh ´ erente de n’importe quelle fonction- ´ nelle faiblement continue de la distribution des variables reponses. Plus particuli ´ erement, nous proposons des ` L- et M-fonctionnelles qui sont des estimateurs fortement coherents de n’importe quelle fonctionnelle con- ´ tinue du parametre de position (par exemple, la m ` ediane). Une m ´ ethode d’ajustement robuste du mod ´ ele de ` regression combin ´ ee aux propri ´ et´ es de robustesse des fonctionnelles de tendance centrale fournissent une ´ methode robuste pour l’estimation du param ´ etre de position. La robustesse de notre proc ` edure est mesur ´ ee´ a l’aide du point de rupture. Nous obtenons aussi la fonction de distribution asymptotique des estimateurs ` du parametre de position. Des suppl ` ements, contenant les d ´ emonstrations des th ´ eor ´ emes, sont disponibles ` en ligne.Fil: Sued, Raquel Mariela. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Cálculo; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Yohai, Victor Jaime. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin