Gérard Antille

A note on optimal designs in weighted polynomial regression for the classical efficiency functions

Abstract In this note we consider the D-optimal design problem for the heteroscedastic polynomial regression model. Karlin and Studden (Ann. Math. Statist. 37 (1966a) 783) found explicit solutions for three types of efficiency functions. We introduce two “new” functions to model the heteroscedastic structure, for which the D-optimal designs can also be found explicitly. The optimal designs have equal masses at the roots of generalized Bessel- and Jacobi-polynomials with complex parameters. It is also demonstrated that there exist no other efficiency functions such that the supporting polynomial of the D-optimal design satisfies a generalized Rodrigues’ formula.

Stability of robust and non-robust principal components analysis

Abstract This paper deals with the stability of robust principal components analysis (PCA) defined through robust estimates of the population covariance matrix as M-estimators or the MVE-estimator. The stability is measured by means of an angular measure between sample principal components and population principal components, the latter being obtained by bootstraping. The studies performed on different data sets show that robust methods do not always improve the stability of PCA. This allows the statistician to choose between robust and nonrobust PCA.

A robust look at the use of regression diagnostics

The need to pay special attention to atypical data in regression analysis is generally accepted. First, because of their excessive influence on the regression results. However, also as emphasized by Gray, the unusual data often provide useful information. Thus, even if robust regression techniques offer a remedy to the fitting problem, the need for regression diagnostics remains. Robust techniques lead to powerful remoteness indicators which, unlike the classical measures based on least squares, are themselves insensitive to atypical data. A re- examination of the two examples discussed by Gray shows that these robust indicators advantageously complement the information obtained with classical influence measures.

Robust and Classical Outlyingness Indicators: A Simulation Study

Robust outlyingness indicators provide a more reliable alternative to least square diagnostics. This paper explains why and illustrates this superiority with a simulation study.

Robust versus classical detection of atypical data

Residuals from a robust fit and robust distances are discussed and used as indicators of outliers or leverage points. The performance of these robust diagnostics are compared to the classical ones by means of a simulation study.

ROBUST LOWER RANK APPROXIMATION OF MATRICES

Mathematical solution of the problem of reduced rank approximation with singular row and column-weights matrices is given. An iteratively reweighted algorithm based on the previous result is proposed to construct low rank approximation of matrices resistant to spurious data. Examples are discussed.