Jorge Adrover

A robust predictive approach for canonical correlation analysis

Canonical correlation analysis (CCA) is a dimension-reduction technique in which two random vectors from high dimensional spaces are reduced to a new pair of low dimensional vectors after applying linear transformations to each of them, retaining as much information as possible. The components of the transformed vectors are called canonical variables. One seeks linear combinations of the original vectors maximizing the correlation subject to the constraint that they are to be uncorrelated with the previous canonical variables within each vector. By these means one actually gets two transformed random vectors of lower dimension whose expected square distance has been minimized subject to have uncorrelated components of unit variance within each vector. Since the closeness between the two transformed vectors is evaluated through a highly sensitive measure to outlying observations as the mean square loss, the linear transformations we are seeking are also affected. In this paper we use a robust univariate dispersion measure (like an M-scale) based on the distance of the transformed vectors to derive robust S-estimators for canonical vectors and correlations. An iterative algorithm is performed by exploiting the existence of efficient algorithms for S-estimation in the context of Principal Component Analysis. Some convergence properties are analyzed for the iterative algorithm. A simulation study is conducted to compare the new procedure with some other robust competitors available in the literature, showing a remarkable performance. We also prove that the proposal is Fisher consistent.

Globally robust inference for the location and simple linear regression models

We define globally robust confidence intervals and p-values for the location and simple linear regression models. The need for robust inference has been noticed and partially addressed in the statistical literature (see for example the book by Barnett and Lewis, Outliers in Statistical Data, Wiley, New York, 1994 and references therein). We construct intervals that are stable in the sense of achieving coverages near the nominal ones even in the presence of outliers and other departures from the parametric model. Moreover, our intervals are informative in the sense of having relatively short lengths. These globally robust confidence intervals constitute an improvement over previous robust intervals which do not take into account the potential bias of the estimates.

Relationships between maximum depth and projection regression estimates

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 equivalent to depth. Two approximate algorithms based on subsampling are proposed to compute the maximum depth estimate: one based on the original definition, and the other on the new criterion. Finally, a Monte Carlo study is performed to compare the least median of squares estimate, the projection estimate, and the two approximate versions of the maximum depth estimate. It is seen that the latter have rather different behaviors, and the best overall performance corresponds to the projection estimate and one of the versions of the depth estimate.

Globally robust confidence intervals for simple linear regression

It is well known that when the data may contain outliers or other departures from the assumed model, classical inference methods can be seriously affected and yield confidence levels much lower than the nominal ones. This paper proposes robust confidence intervals and tests for the parameters of the simple linear regression model that maintain their coverage and significance level, respectively, over whole contamination neighbourhoods. This approach can be used with any consistent regression estimator for which maximum bias curves are tabulated, and thus it is more widely applicable than previous proposals in the literature. Although the results regarding the coverage level of these confidence intervals are asymptotic in nature, simulation studies suggest that these robust inference procedures work well for small samples, and compare very favourably with earlier proposals in the literature.

Approximate τ—Estimates for Linear Regression Based on Subsampling of Elemental Sets

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 minimum of the τ-scale having the same asymptotic distribution and, with high probability, the same breakdown point that the global minimum.

Efficiency of MM- and [tau]-estimates for finite sample size

Suppose that the relative efficiency of a regression estimate with respect to the least squares estimate is measured using a robust scale. Then, it is shown that in the case of normal errors and a finite sample size, it is possible to find MM- and [tau]-estimates which combine high efficiency and high breakdown-point.

A new projection estimate for multivariate location with minimax bias

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 mass contaminations (the worst situation for the projection median estimator). Moreover, it achieves the lowest bound for an equivariant estimate for point mass contaminations.

Quantile estimation for a selected normal population

In this paper simultaneous redescending M-estimates for scale and regression parameters are properly defined. The breakdown point of this estimates is derived. It is proved that these estimates are asynlptoticaliy normal and their cavariance matrix is obtained. These results show that simultaneous redescending M-estimates may combine high breakdown point and high asymptotic efficiency Under normal errors.