We derive the asymptotic distribution ofthe maximal depth regression estimator recently proposed in Rousseeuw and Hubert. The estimator is obtained by maximizing a projection-based depth and the limiting distribution is characterized through a max–min operation of a continuous process. The same techniques can be used to obtain the limiting distribution of some other depth estimators including Tukey’s deepest point based on half-space depth. Results for the special case of two-dimensional problems have been available, but the earlier arguments have relied on some special geometric properties in the low-dimensional space. This paper completes the extension to higher dimensions for both regression and multivariate location models.
"Asymptotic distributions of the maximal depth estimators for regression and multivariate location." Ann. Statist. 27 (5) 1616 - 1637, October 1999. https://doi.org/10.1214/aos/1017939144