Victor J. Yohai

University of Buenos Aires | UBA · Department of Mathematics (FCEN)

Given a high-dimensional vector of time series, we define a class of robust forecasting procedures based on robust one-sided dynamic principal components. Peña et al. (J Am Stat Assoc 114(528):1683–1694, 2019) defined one-sided dynamic principal components as linear combinations of the present and past values of the series with optimal reconstructi...

We propose a class of Fisher-consistent robust estimators for mixture models. These estimators are then used to build a robust model-based clustering procedure. We study in detail the case of multivariate Gaussian mixtures and propose an algorithm, similar to the EM algorithm, to compute the proposed estimators and build the robust clusters. An ext...

Let \(F_{\theta }\) be a family of distributions with support on the set of nonnegative integers \(Z_{0}\). In this paper we derive the M-estimators with smallest gross error sensitivity (GES). We start by defining the uniform median of a distribution F with support on \(Z_{0}\) (umed(F)) as the median of \(x+u,\) where x and u are independent vari...

We propose a new class of robust and Fisher-consistent estimators for mixture models. These estimators can be used to construct robust model-based clustering procedures. We study in detail the case of multivariate normal mixtures and propose a procedure that uses S estimators of multivariate location and scatter. We develop an algorithm to compute...

In nonparametric regression contexts, when the number of covariables is large, we face the curse of dimensionality. One way to deal with this problem when the sample is not large enough is using a reduced number of linear combinations of the explanatory variables that contain most of the information about the response variable. This leads to the so...

We present the sparse estimation of one-sided dynamic principal components (ODPCs) to forecast high-dimensional time series. The forecast can be made directly with the ODPCs or by using them as estimates of the factors in a generalized dynamic factor model. It is shown that a large reduction in the number of parameters estimated for the ODPCs can b...

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.

gdpc is an R package for the computation of the generalized dynamic principal components proposed in Peña and Yohai (2016). In this paper, we briefly introduce the problem of dynamical principal components, propose a solution based on a reconstruction criteria and present an automatic procedure to compute the optimal reconstruction. This solution c...

Let F_{{\theta}} be a family of distributions with support on the set of nonnegative integers Z_0. In this paper we derive the M-estimators with smallest gross error sensitivity (GES). We start by defining the uniform median of a distribution F with support on Z_0 (umed(F)) as the median of x+u, where x and u are independent variables with distribu...

K means is a popular non-parametric clustering procedure introduced by Steinhaus (1956) and further developed by MacQueen (1967). It is known, however, that K means does not perform well in the presence of outliers. Cuesta-Albertos et al (1997) introduced a robust alternative, trimmed K means, which can be tuned to be robust or efficient, but canno...

Generalized Linear Models are routinely used in data analysis. Classical estimators are based on the maximum likelihood principle and it is well known that the presence of outliers can have a large impact on them. Several robust procedures have been presented in the literature, being redescending M-estimators the most widely accepted. Based on non-...

This chapter focuses on time series in discrete time. It expresses that the time series is either stationary in some sense or may be reduced to stationarity by a combination of elementary differencing operations and regression trend removal. Two types of stationarity are in common use, second‐order stationarity and strict stationarity. A strictly s...

This chapter discusses the estimation of the parameters of linear regression models. It reviews the main properties of least squares (LS) for multiple regression. The most popular way to deal with regression outliers is to use LS and try to find the influential observations. After they have been identified, a decision must be taken, for example mod...

This chapter considers more general situations in which the regressors affect the distribution function of the response variable (y). In doing so, it first considers the situation when y is a 0‐1 variable. The binary response regression model is included in a more general class called generalized linear models (GLMs). Next, the chapter discusses th...

This chapter establishes a general family of estimators that contains the mean and the median as special cases. There are several methods available for computing M‐estimators of location and/or scale. In principle one could use any of the general methods for equation solving such as the Newton‐Raphson algorithm, but methods based on derivatives may...

Computing M‐estimators involves function minimization and/or solving nonlinear equations. General methods based on derivatives ‐ like the Newton‐Raphson procedure for solving equations ‐ are widely available, but they are inadequate for this specific type of problem. This chapter considers some details of the iterative algorithms used to compute M‐...

This chapter describes the datasets used in this book. It also mentions the companion websites for these datasets. Data collected in a broad range of applications frequently contain one or more atypical observations, known as outliers; that is, observations that are well‐separated from the majority or "bulk" of the data, or in some way deviate from...

In order to compare the performances of different estimators, and also to obtain confidence intervals for the parameters, one needs their distributions. Explicit expressions exist in some simple cases, such as sample quantiles, which include the median, but even these are in general intractable. It will be necessary to resort to approximating their...

Multivariate analysis deals with situations in which several variables are measured on each experimental unit. In most cases of interest it is known or assumed that some form of relationship exists among the variables, and hence that considering each of them separately would entail a loss of information. Some possible goals of the analysis are: red...

For purposes of plotting and comparing sensitivity curves across sample sizes, it is convenient to use standardized sensitivity curves. The influence function (IF) of an estimator is an asymptotic version of its sensitivity curve. The breakdown point (BP) for each type of estimator has to be treated separately. It is easy to find estimators with hi...

This chapter deals with the case of random predictors and focuses on how to obtain good initial values for redescending M‐estimators. It presents an example that shows the failure of a monotone M‐estimator when X is random and there is a single atypical observation. The chapter briefly discusses the properties of a linear model with random X. It de...

Highly robust and efficient estimators for generalized linear models with a dispersion parameter are proposed. The estimators are based on three steps. In the first step, the maximum rank correlation estimator is used to consistently estimate the slopes up to a scale factor. The scale factor, the intercept, and the dispersion parameter are robustly...

Comments on the “monitoring” method and its relationships with other robust estimation methods.

In this paper we propose a new procedure to estimate the distribution of a variable y when there are missing data. To compensate the presence of missing responses it is assumed that a covariate vector x is observed and that y and x are related by means of a semi-parametric regression model. Observed residuals are combined with predicted values to e...

Generalized Linear Models are routinely used in data analysis. The classical procedures for estimation are based on Maximum Likelihood and it is well known that the presence of outliers can have a large impact on this estimator. Robust procedures are presented in the literature but they need a robust initial estimate in order to be computed. This i...

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. Previous definitions of dynamic principal components depend on past and future values of the series. For this reason, they are not appropriate for forecastin...

Doubly protected estimators are widely used for estimating the population mean of an outcome Y from a sample where the response is missing in some individuals. To compensate for the missing responses, a vector X of covariates is observed at each individual, and the missing mechanism is assumed to be independent of the response, conditioned on X (mi...

Highly robust and efficient estimators for the generalized linear model with a dispersion parameter are proposed. The estimators are based on three steps. In the first step the maximum rank correlation estimator is used to consistently estimate the slopes up to a scale factor. In the second step, the scale factor, the intercept, and the dispersion...

The generalized log-gamma (GLG) model is a very flexible family of distributions to analyze datasets in many different areas of science and technology. In this paper, we propose estimators which are simultaneously highly robust and highly efficient for the parameters of a GLG distribution in the presence of censoring. We also introduced estimators...

Real data may contain both cellwise outliers and casewise outliers. There is a vast literature on robust estimation for casewise outliers, but only a scant literature for cellwise outliers and almost none for both types of outliers. Estimation of multivariate location and scatter matrix is a corner stone in multivariate data analysis. A two-step ap...

Several equivariant estimators of multivariate location and scatter are studied, which are highly robust, have a controllable finite-sample efficiency and are computationally feasible in large dimensions. The most frequently employed estimators are not quite satisfactory in this respect. The Minimum Volume Ellipsoid (MVE) and the Minimum Covariance...

Classical methods in multivariate analysis require the estimation of means and covariance matrices. Although the sample mean and covariance matrix are optimal estimates of multivariate location and scatter when the data are multivariate normal, a small fraction of atypical points in the data (outliers) suffices to drastically alter them. This artic...

robustloggamma is an R package for robust estimation and inference in the generalized loggamma model. We briefly introduce the model, the estimation procedures and the computational algorithms. Then, we illustrate the use of the package with the help of a real data set.

Penalized regression estimators are a popular tool for the analysis of sparse and high-dimensional data sets. However, penalized regression estimators defined using an unbounded loss function can be very sensitive to the presence of outlying observations, especially high leverage outliers. Moreover, it can be particularly challenging to detect outl...

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 procedur...

Multivariate location and scatter matrix estimation is a cornerstone in multivariate data analysis. We consider this problem when the data may contain independent cellwise and casewise outliers. Flat data sets with a large number of variables and a relatively small number of cases are common place in modern statistical applications. In these cases...

We thank the discussants, the referees, and the associate editor for their stimulating discussions and helpful remarks. We thank the editor for giving us the opportunity to discuss our paper in this journal. The rejoinder is organized in several sections, which address the main points raised by the discussants.

We deal with the equivariant estimation of scatter and location for p-dimensional data, giving emphasis to scatter. It it important that the estimators possess both a high efficiency for normal data and a high resistance to outliers, that is, a low bias under contamination. The most frequently employed estimators are not quite satisfactory in this...

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-estimato...

ANOVA tests are the standard tests to compare nested linear models fitted by least squares. These tests are equivalent to likelihood ratio tests, so they have high power. However, least squares estimators are very vulnerable to outliers in the data, and thus the related ANOVA type tests are also extremely sensitive to outliers. Therefore, robust es...

The Classical Tukey-Huber Contamination Model (CCM) is a usual framework to describe the mechanism of outliers generation in robust statistics. In a data set with $n$ observations and $p$ variables, under the CCM, an outlier is a unit, even if only one or few values are corrupted. Classical robust procedures were designed to cope with this setting...

We propose a time domain approach to define dynamic principal components (DPC) using a reconstruction of the original series criterion. This approach to define DPC was introduced by Brillinger, who gave a very elegant theoretical solution in the stationary case using the cross spectrum. Our procedure can be applied under more general conditions inc...

In this paper we propose a family of robust estimators for generalized linear models. The basic idea is to use an M-estimator after applying a variance stabilizing transformation to the response. We show the consistency and asymptotic normality of these estimators. We also obtain a lower bound for their breakdown point. A Monte Carlo study shows th...

In this paper we propose a family of robust estimators for generalized linear models. The basic idea is to use an M-estimator after applying a variance stabilizing transformation to the response. We show the consistency and asymptotic normality of these estimators. We also obtain a lower bound for their breakdown point. A Monte Carlo study shows th...

We propose robust estimators of the generalized log-gamma distribution and, more generally, of location-shape-scale families of distributions. A (weighted) Qτ estimator minimizes a τ scale of the differences between empirical and theoretical quantiles. It is n 1/2 consistent; unfortunately, it is not asymptotically normal and, therefore, inconvenie...

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 tau-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 f...

Two robust estimators of a matrix-valued location parameter are introduced and discussed. Each is the average of the members of a subsample–typically of covariance or cross-spectrum matrices–with the subsample chosen to minimize a function of its average. In one case this function is the Kullback–Leibler discrimination information loss incurred whe...

All known approaches to nonlinear principal components are based on minimizing a quadratic loss, which makes them sensitive to data contamination. A predictive approach in which a spline curve is fit minimizing a residual M-scale is proposed for this problem. For a p-dimensional random sample x i (i=1,…,n) the method finds a function h:R→R p and a...

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...

Two main issues regarding data quality are data contamination (outliers) and data completion (missing data). These two problems have attracted much attention and research but surprisingly, they are seldom considered together. Popular robust methods such as S-estimators of multivariate location and scatter offer protection against outliers but canno...

In this paper we propose a family of robust estimates for isotonic regression: isotonic M-estimators. We show that their asymptotic distribution is, up to an scalar factor, the same as that of Brunk's classical isotonic estimator. We also derive the influence function and the breakdown point of these estimates. Finally we perform a Monte Carlo stud...

In this paper we show that approximate τ-estimates for the linear model, computed by the algorithm based on subsampling of elemental subsets, are consistent and with high probability have the same breakdown point that the exactτ-estimate. Then, if these estimates are used as initial values, the reweighted least squares algorithm yields a local mini...

Robust estimators for accelerated failure time models with asymmetric (or symmetric) error distribution and censored observations are proposed. It is assumed that the error model belongs to a log-location-scale family of distributions and that the mean response is the parameter of interest. Since scale is a main component of mean, scale is not trea...

Existing methods for functional regression are based on an L2 norm of the residuals and are therefore sensitive to atypical observations, which may aect the predictive power and/or the smoothness of the resulting estimate. We propose a robust version of the spline-based estimate pro- posed by Crambes, Kneip and Sarda (Ann. Statist. 2009), which has...

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 pp remains fixed while the number of observations nn tends to infinity, which means assuming that the ratio p...

Optimal robust M-estimates of a multidimensional parameter are described using Hampel’s infinitesimal approach. The optimal estimates are derived by minimizing a measure of efficiency under the model, subject to a bounded measure of infinitesimal robustness. To this purpose we define measures of efficiency and infinitesimal sensitivity based on the...

The maximum asymptotic bias of an estimator is a global robustness measure of its performance. The projection median estimator for multivariate location shows a remarkable behavior regarding asymptotic bias. In this paper we consider a modification of the projection median estimator which renders an estimate with better bias performance for point m...

This paper deals with the Fisher-consistency, weak continuity and differentiability of estimating functionals corresponding to a class of both linear and nonlinear regression high breakdown M estimates, which includes S and MM estimates. A restricted type of differentiability, called weak differentiability, is defined, which suffices to prove the a...

This paper deals with the weak continuity, Fisher-consistency and differentiability of estimating functionals corresponding to a class of both linear and nonlinear regression high breakdown M estimates, which includes \ S and MM estimates. A restricted type of differentiability, called weak differentiability, is defined, which suffices to prove the...

During the preimplantation phase of pregnancy the endometrial stroma differentiates into decidua, a process that implies numerous morphological changes and is an example of physiological transdifferentiation. Here we show that UIII rat endometrial stromal cells cultured in the presence of calf serum acquired morphological features of decidual cells...

It is shown that the sliced inverse regression procedure proposed by Li corresponds to the maximum likelihood estimate where the observations in each slice are samples of multivariate normal distributions with means in an affine manifold.

This paper introduces a new class of robust estimates for ARMA models. They are M-estimates, but the residuals are computed so the effect of one outlier is limited to the period where it occurs. These estimates are closely related to those based on a robust filter, but they have two important advantages: they are consistent and the asymptotic theor...

We investigate the performance of robust estimates of multivariate location under nonstandard data contamination models such as componentwise outliers (i.e., contamination in each variable is independent from the other variables). This model brings up a possible new source of statistical error that we call "propagation of outliers." This source of...

Nonlinear regression problems can often be reduced to linearity by transforming the response variable (e.g., using the Box-Cox family of transformations). The classic estimates of the parameter defining the transformation as well as of the regression coefficients are based on the maximum likelihood criterion, assuming homoscedastic normal errors fo...

New robust estimates for variance components are introduced. Two simple models are considered: the balanced one-way classification model with a random factor and the balanced mixed model with one random factor and one fixed factor. However, the method of estimation we proposed can be extended to more complex models. The new method of estimation we...

In this paper we present two robust estimates for GARCH models. The first is defined by the minimization of a conveniently modified likelihood and the second is similarly defined, but includes an additional mechanism for restricting the propagation of the effect of one outlier on the next estimated conditional variances. We study the asymptotic pro...

We define two measures of the performance of an estimating functional T of a multi-dimensional parameter, based on the Kullback-Leibler (KL) divergence. The first one is the KL sensitivity which measures the degree of robustness of the estimate under infinitesimal outlier contamination and the second one is the KL efficiency, which measures the asy...

In this paper we propose a robust method to approximate an np data matrix with one of given rank q. The method is based on Yohai's (1987) regression MM estimates. It is intended to be resistant against the existence of both atypical rows and of scattered atypical cells, and to be able to cope with missing values. We propose an algorithm based on al...

In this paper, we propose a class of high breakdown point estimators for the linear regression model when the response variable contains censored observations. These estimators are robust against high-leverage outliers and they generalize the LMS (least median of squares), S, MM and $\tau$-estimators for linear regression. An important contribution...

We show, using a Monte Carlo study, that MM-estimates with projection estimates as starting point of an iterative weighted least squares algorithm, behave more robustly than MM-estimates starting at an S-estimate and similar Gaussian efficiency. Moreover the former have a robustness behavior close to the P-estimates with an additional advantage: th...

Classical methods in multivariate analysis require the estimation of means and covariance matrices. Although the sample mean and covariance matrix are optimal estimates of multivariate location and scatter when the data are multivariate normal, a small fraction of atypical points in the data (outliers) suffices to drastically alter them. This artic...

1~l the Least Squares Estimator (LSE) of 8 is optimal when H is normal, it is nonrobust in the sense that arbitrarily small departures of H from normality may cause arbitrarily large asymptotic variances and/or biases of the estimator. A first step towards robustness is given by the "classical M-estimators" defined as solutions 8" of equations of t...

We introduce a class of robust estimates for multivariate linear models. The regression coefficients and the covariance matrix of the errors are estimated simultaneously by minimizing the determinant of the covariance matrix estimate, subject to a constraint on a robust scale of the Mahalanobis norms of the residuals. By choosing a [tau]-estimate a...

Equivariant high-breakdown point regression estimates are computationally expen-sive, and the corresponding algorithms become unfeasible for moderately large number of regressors. One important advance to improve the computational speed of one such estima-tor is the fast-LTS algorithm. This article proposes an analogous algorithm for computing S-es...

Time series outliers and their impactClassical estimates for AR modelsClassical estimates for ARMA modelsM-estimates of ARMA modelsGeneralized M-estimatesRobust AR estimation using robust filtersRobust model identificationRobust ARMA model estimation using robust filtersARIMA and SARIMA modelsDetecting time series outliers and level shiftsRobustnes...

We present a random coefficient regression model in which a response is linearly related to some explanatory variables with random coefficients following a Dirichlet distribution. These coefficients can be interpreted as weights because they are nonnegative and add up to one. The proposed estimation procedure combines iteratively reweighted least s...

Response transformations are a popular approach to adapt data to a linear regression model. The regression coefficients, as well as the parameter defining the transformation, are often estimated by maximum likelihood assuming homoscedastic normal errors. Unfortunately, consistency to the true parameters holds only if the assumptions of normality an...

The authors propose a new class of robust estimators for the parameters of a regression model in which the distribution of the error terms belongs to a class of exponential families including the log-gamma distribution. These estimates, which are a natural extension of the MM-estimates for ordinary regression, may combine simultaneously high asympt...

In this work we study the procedure of dimension reduction for multivariate observations known as Sliced Inverse Regression (SIR) presented by K. C. Li (1991). We prove that the algorithm developed by Li (1991) to solve the problem of sliced inverse regression provides the same results as those obtained by the maximum likelihood method of estimatin...

We consider the problem of constructing robust nonparametric confidence intervals and tests of hypothesis for the median when the data distribution is unknown and the data may contain a small fraction of contamination. We propose a modification of the sign test (and its associated confidence interval) which attains the nominal significance level (p...

We consider robust estimators for the linear regression model with asymmetric (or symmetric) error distribution. We assume that the error model belongs to a location-scale family of distributions. Since in the asymmetric case the mean response is very often the parameter of interest and scale is a main component of mean, we do not assume that scale...

The regression quantile estimate introduced by Koenker and Bassett in 1978 may not be robust when the predictors contain leverage points. We define estimates which are free of this drawback, and furthermore attain the maximum breakdown point for this problem. Simulations show them to behave generally better than competing robust quantile estimates.

The normal quantile–quantile (Q–Q) plot of residuals is a popular diagnostic tool for ordinary linear regression with normal errors. However, for some generalized linear regression models, the distribution of deviance residuals may be very far from normality,and therefore the corresponding normal Q–Q plots may be misleading to check model adequacy....

The use of the Box-Cox family of transformations is a popular approach to make data behave according to a linear regression model. The regression coefficients, as well as the parameter λ defining the transformation, are generally estimated by maximum likelihood, assuming homoscedastic nor-mal errors. These estimates are nonrobust; in addition, cons...

P. Rousseeuw [Math. Stat. Applications, Proc. 4th Pannonian Symp. Math. Stat., 283–297 (1985; Zbl 0609.62054)] introduced the Minimum Volume Ellipsoid (MVE) estimates of covariance matrices and multivariate location. These estimates, which are broadly used, are affine equivariant and have high breakdown point. C. Croux et al. [J. Nonparametric Stat...

In this paper, we present two robust estimates for ARCH(p) models: A - and filtered A-estimates. These are defined by the minimization of conveniently robustified likelihood functions. The robustification is achieved by replacing the mean square error of the standardized observations with the square of a robust A-scale estimate in the reduced form...

A new class of estimates for the linear model is introduced. These estimates, that we call C-estimates, are defined as a convex combination of a high breakdown point estimate, , and any other estimate, . We prove that C-estimates retain the global robustness properties of and inherit the local robustness behavior and the asymptotic distribution of...

In this paper we study the maximum asymptotic bias of the projection estimate for multivariate location based on univariate estimates of location and dispersion. In particular we study the projection estimate that uses the median and median absolute deviation about the median (MAD) as univariate location and dispersion estimates respectively. This...

This article deals with the relationships between regression estimates based on projections (Ann. Statist. 21 (1993) 965) and on maximum depth (J. Amer. Statist. Assoc. 94 (1999) 388), which are shown to share the same basic idea. The maximum asymptotic bias of the latter is derived. A measure of residual smallness is defined, which turns out to be...

This paper introduces a new class of robust estimators for the linear regression model. They are weighted least squares estimators, with weights adaptively computed using the empirical distribution of the residuals of an initial robust estimator. It is shown that under certain general conditions the asymptotic breakdown points of the proposed estim...

Bustos and Yohai proposed a class of robust estimates for autoregressive moving-average (ARMA) models based on residual autocovariances (RA estimates). In this paper an affine equivariant generalization of the RA estimates for vector ARMA processes is given. These estimates are asymptotically normal and, when the innovations have an elliptical dist...

We find a family of M-estimates of regression with the following minimax bias property: They minimize the asymptotic variance at the central model subject to a bound on the maximum bias over contamination neighborhoods. For the case of multivariate normal data, the optimal ψ-functions associated with the optimal estimates are numerically computed....

A diagnostic procedure for detecting additive and innovation outliers as well as level shifts in a regression model with ARIMA errors is introduced. The procedure is based on a robust estimate of the model parameters and on innovation residuals computed by means of robust filtering. A Monte Carlo study shows that, when there is a large proportion o...

The probability of Trypanosoma cruzi transmission to opossums by independent events of predation and fecal contamination during feeding ("biting") with positive Triatoma infestans was estimated. Negative female opossums were challenged for 23 hr with 10 infected third and fourth instars of T. infestans, and tests for positivity for T. cruzi by xeno...

We find optimal robust estimates for the location parameter of n independent measurements from a common distribution F that belongs to a contamination neighborhood of a normal distribution. We follow an asymptotic minimax approach similar to Huber's but work with full neighborhoods of the central parametric model including nonsymmetric distribution...