The Distribution of Robust Distances

Mahalanobis-type distances in which the shape matrix is derived from a consistent, high-breakdown robust multivariate location and scale estimator have an asymptotic chisquared distribution as is the case with those derived from the ordinary covariance matrix. For example, Rousseeuw’s minimum covariance determinant (MCD) is a robust estimator with a high breakdown. However, even in quite large samples, the chi-squared approximation to the distances of the sample data from the MCD center with respect to the MCD shape is poor. We provide an improved F approximation that gives accurate outlier rejection points for various sample sizes.