On Weighted Multivariate Sign Functions

Multivariate sign functions are often used for robust estimation and inference. We propose using data dependent weights in association with such functions. The proposed weighted sign functions retain desirable robustness properties, while significantly improving efficiency in estimation and inference compared to unweighted multivariate sign-based methods. Using weighted signs, we demonstrate methods of robust location estimation and robust principal component analysis. We extend the scope of using robust multivariate methods to include robust sufficient dimension reduction and functional outlier detection. Several numerical studies and real data applications demonstrate the efficacy of the proposed methodology.Comment: Keywords: Multivariate sign, Principal component analysis, Data depth, Sufficient dimension reductio

Robust canonical correlations: a comparative study.

Several approaches for robust canonical correlation analysis will be presented and discussed. A first method is based on the definition of canonical correlation analysis as looking for linear combinations of two sets of variables having maximal (robust) correlation. A second method is based on alternating robust regressions. These methods are discussed in detail and compared with the more traditional approach to robust canonical correlation via covariance matrix estimates. A simulation study compares the performance of the different estimators under several kinds of sampling schemes. Robustness is studied as well by breakdown plots.Alternating regression; Canonical correlations; Correlation measures; Projection-pursuit; Robust covariance estimation; Robust regression; Robustness;

Resistant estimates for high dimensional and functional data based on random projections

We herein propose a new robust estimation method based on random projections that is adaptive and, automatically produces a robust estimate, while enabling easy computations for high or infinite dimensional data. Under some restricted contamination models, the procedure is robust and attains full efficiency. We tested the method using both simulated and real data.Comment: 24 pages, 6 figure

The Gaussian rank correlation estimator: Robustness properties.

The Gaussian rank correlation equals the usual correlation coefficient computed from the normal scores of the data. Although its influence function is unbounded, it still has attractive robustness properties. In particular, its breakdown point is above 12%. Moreover, the estimator is consistent and asymptotically efficient at the normal distribution. The correlation matrix based on the Gaussian rank correlation is always positive semidefinite, and very easy to compute, also in high dimensions. A simulation study confirms the good efficiency and robustness properties of the proposed estimator with respect to the popular Kendall and Spearman correlation measures. In the empirical application, we show how it can be used for multivariate outlier detection based on robust principal component analysis.Breakdown; Correlation; Efficiency; Robustness; Van der Waerden;

Partial robust M-regression.

Partial Least Squares (PLS) is a standard statistical method in chemometrics. It can be considered as an incomplete, or 'partial', version of the Least Squares estimator of regression, applicable when high or perfect multicollinearity is present in the predictor variables. The Least Squares estimator is well-known to be an optimal estimator for regression, but only when the error terms are normally distributed. In the absence of normality, and in particular when outliers are in the data set, other more robust regression estimators have better properties. In this paper a 'partial' version of M-regression estimators will be defined. If an appropriate weighting scheme is chosen, partial M-estimators become entirely robust to any type of outlying points, and are called Partial Robust M-estimators. It is shown that partial robust M-regression outperforms existing methods for robust PLS regression in terms of statistical precision and computational speed, while keeping good robustness properties. The method is applied to a data set consisting of EPXMA spectra of archaeological glass vessels. This data set contains several outliers, and the advantages of partial robust M-regression are illustrated. Applying partial robust M-regression yields much smaller prediction errors for noisy calibration samples than PLS. On the other hand, if the data follow perfectly well a normal model, the loss in efficiency to be paid for is very small.Advantages; Applications; Calibration; Data; Distribution; Efficiency; Estimator; Least-squares; M-estimators; Methods; Model; Optimal; Ordinary least squares; Outliers; Partial least squares; Precision; Prediction; Projection-pursuit; Regression; Robust regression; Robustness; Simulation; Spectometric quantization; Squares; Studies; Variables; Yield;