Asymptotic distributions of the maximal depth estimators for regression and multivariate location

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Abstract
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.