An Extension of a Method of Hardin and Rocke , with an Application to Multivariate Outlier Detection via the IRMCD Method of Cerioli

Hardin and Rocke investigated the distribution of the robust Mahalanobis squared distance (RSD) computed using the minimum covariance determinant (MCD) estimator. They showed that the distribution of RSDs for outlying observations not part of the MCD subset is well-approximated by an F distribution. They developed a methodology to adjust an asymptotic formula for the degrees of freedom parameters of this F distribution to provide correct parameter values in small-to-moderate samples. This methodology was developed for the maximum breakdown point version of the MCD, which is based on approximately half of the observations. Whether the approximation remains accurate for the MCD using larger subsets of the data is an open question. We show that their approximation works quite well for the more general MCD, but can be noticeably inaccurate for sample sizes less than 250 and when the MCD estimate uses nearly all of the observations. Motivated by the desire to apply RSD-based outlier detection tests to financial asset return and factor exposure data sets whose typical sample sizes are smaller than 250, we develop a more general correction procedure that is accurate across a wider range of sample sizes and MCD subset sizes than the Hardin and Rocke approach. We use our approach to extend Cerioli’s IRMCD procedure for accurate RSD-based outlier tests to arbitrary MCD subset sizes.