Fast Algorithms for Computing High Breakdown Covariance Matrices with Missing Data

Abstract

Robust estimation of covariance matrices when some of the data at hand are missing is an important problem. It has been studied by Little and Smith (1987) and more recently by Cheng and Victoria-Feser (2002). The latter propose the use of high breakdown estimators and so-called hybrid algorithms (see e.g. Woodruff and Rocke 1994). In particular, the minimum volume ellipsoid of Rousseeuw (1984) is adapted to the case of missing data. To compute it, they use (a modified version of) the forward search algorithm (see e.g. Atkinson 1994). In this paper, we propose to use instead a modification of the C-step algorithm proposed by Rousseeuw and Van Driessen (1999) which is actually a lot faster. We also adapt the orthogonalized Gnanadesikan-Kettering (OGK) estimator proposed by Maronna and Zamar (2002) to the case of missing data and use it as a starting point for an adapted S-estimator. Moreover, we conduct a simulation study to compare different robust estimators in terms of their efficiency and breakdown and use them to analyse real datasets.