Influence function analysis applied to partial least squares

In this work we present the empirical influence functions for the covariances (eigenvalues) and directions (eigenvectors) of partial least squares under the constraint of uncorrelated components. We apply the results to several data sets and provide advice for using these tools in practice.     

Nonparametric predictive pairwise comparison for real-valued data with terminated tails

This paper presents a statistical method for comparison of two groups of real-valued data, based on nonparametric predictive inference (NPI), with the tails of the data possibly terminated, leading to small values being left-censored and large values being right-censored. Such tails termination can occur due to several reasons, including limits of detection, consideration of outliers, and specific designs of experiments. NPI is a statistical approach based on few assumptions, with inferences strongly based on data and with uncertainty quantified via lower and upper probabilities. We present NPI lower and upper probabilities for the event that the value of a future observation from one group is less than the value of a future observation from the other group, and we discuss several special cases that relate to well-known statistical problems.     

Duality results and proximal solutions of the Huber M-estimator problem

We investigate the interest of solving the Huber M-estimator problem by a proximal approach combined with duality theory. Three different duality schemes are developed. The first one which only deals with estimator determination yields useful information on the geometrical structure of the set of optimal solutions. The second scheme links together estimator determination and outliers detection while the third one only focuses on outliers separation. We show that these three duality schemes can be solved by the partial inverse method, i.e., a special instance of the basic proximal point algorithm, which leads to very simple updating rules. This method which is always globally convergent enjoys nice stability properties and permits parallel computations.     

Diagnostic Plots for Robust Multivariate Methods

Robust techniques for multivariate statistical methods—such as principal component analysis, canonical correlation analysis, and factor analysis—have been recently constructed. In contrast to the classical approach, these robust techniques are able to resist the effect of outliers. However, there does not yet exist a graphical tool to identify in a comprehensive way the data points that do not obey the model assumptions. Our goal is to construct such graphics based on empirical influence functions. These graphics not only detect the influential points but also classify the observations according to their robust distances. In this way the observations are divided into four different classes which are regular points, nonoutlying influential points, influential outliers, and noninfluential outliers. We thus gain additional insight in the data by detecting different types of deviating observations. Some real data examples will be given to show how these plots can be used in practice.