Ricardo Fraiman

Robust principal component analysis for functional data

A method for exploring the structure of populations of complex objects, such as images, is considered. The objects are summarized by feature vectors. The statistical backbone is Principal Component Analysis in the space of feature vectors. Visual insights come from representing the results in the original data space. In an ophthalmological example, endemic outliers motivate the development of a bounded influence approach to PCA.

Kernel-based functional principal components (

In this paper, we propose kernel-based smooth estimates of the functional principal components when data are continuous trajectories of stochastic processes. Strong consistency and the asymptotic distribution are derived under mild conditions.

Multivariate L-estimation

In one dimension, order statistics and ranks are widely used because they form a basis for distribution free tests and some robust estimation procedures. In more than one dimension, the concept of order statistics and ranks is not clear and several definitions have been proposed in the last years. The proposed definitions are based on different concepts of depth. In this paper, we define a new notion of order statistics and ranks for multivariate data based on density estimation. The resulting ranks are invariant under affinc transformations and asymptotically distribution free. We use the corresponding order statistics to define a class of multivariate estimators of location that can be regarded as multivariate L-estimators. Under mild assumptions on the underlying distribution, we show the asymptotic normality of the estimators. A modification of the proposed estimates results in a high breakdown point procedure that can deal with patches of outliers. The main idea is to order the observations according to their likelihoodf(X1),...,f(Xn). If the densityf happens to be cllipsoidal, the above ranking is similar to the rankings that are derived from the various notions of depth. We propose to define a ranking based on a kernel estimate of the densityf. One advantage of estimating the likelihoods is that the underlying distribution does not need to have a density. In addition, because the approximate likelihoods are only used to rank the observations, they can be derived from a density estimate using a fixed bandwidth. This fixed bandwidth overcomes the curse of dimensionality that typically plagues density estimation in high dimension.

Asymptotic Behaviour of the Estimates Based on Residual Autocovariances for ARMA Models

In a recent paper Bustos and Yohai introduce the class of estimates based on residual auto-covariances (RA-estimates) for the parameters of an ARMA model. They show using a Monte Carlo study that this class contains estimates which are highly efficient when the observations correspond, to a perfectly observed Gaussian ARMA model and robust under the presence of outliers. In this paper we show the consistency and asymptotic normality of a class of estimates containing the RA-estimates.

General m-esttmators and applications to bounded influence estimation for non-linear regression

In this paper, we study the M-estimators in the case that λF:(β)=EF:(φ(Z,β))=0 has more than one solution, We show that the numerical iterative procedures converge and that the resulting estimators are consistent and asymptotically normal. We apply them to the non-linear regression models, and then, we find an optimal M-estimate among those that have bounded gross error sensitivity.

Chain rule density estimates

In this paper, we propose a new family of density estimates closely related to the nearest neighbor estimates introduced by Loftsgaarden and Quesenberry. An optimal estimator, wit respect to the asymptotic mean square error, is obtained for a given distribution.