DISCUSSION PAPERS IN STATISTICS AND ECONOMETRICS SEMINAR OF ECONOMIC AND SOCIAL STATISTICS UNIVERSITY OF COLOGNE No. 3/07 Distribution-Free Shape Matrix Estimation for Incomplete Data

Many different robust estimation approaches for the covariance matrix or, say, the shape matrix of multivariate data have been established until today. Tyler’s M-estimator has been recognized as the ‘most robust’ M-estimator for elliptically symmetric distributed data. Essentially, given the class of elliptically symmetric distributions it is a distribution-free MLestimator. We show that this property holds even if the data are generalized elliptically distributed and we extend the distribution-free estimation approach to the case of incomplete data. It turns out that the resulting spectral estimator is still a robust ML-estimator. We present a fast algorithm for calculating the spectral estimate which works well also for highdimensional data. Further, we provide both an empirical and a simulation study. For the empirical study we analyze daily returns of stocks listed by the S&P 500 stock index. We show that the spectral estimator leads to robust estimates of the principal components if the stock market crashes. The simulation study covers the complete-data as well as the incompletedata case with clean and contaminated data. It reveals that the spectral estimator is extremely robust against heavy-tailed or contaminated data whereas the loss of efficiency in case of uncontaminated and light-tailed data seems to be quite negligible.