Finite sample tail behavior of the multivariate trimmed mean based on Tukey-Donoho halfspace depth

Finite sample tail behavior of the Tukey-Donoho halfspace depth based multivariate trimmed mean is investigated with respect to a tail performance measure. It turns out that the tails of the sampling distribution of the alpha -depth-trimmed mean approach zero at least [alphan] times as fast as the tails of the underlying population distribution and could be n - [alphan] + 1 times as fast. In addition, there is an intimate relationship among the tail behavior, the halfspace depth, and the finite sample breakdown point of the estimator. It is shown that the lower tail performance bound of the depth based trimmed mean is essentially the same as its halfspace depth and the breakdown point. This finding offers anew insight into the notion of the halfspace depth and extends the important role of the tail behavior as a quantitative assessment of robustness in the regression (He, Jureckova, Koenker and Portnoy (1990)) and the univariate location settings (Jureckova (1981)) to the multivariate location setting. AMS 1991 Classification: Primary 62F35; Secondary 62G35, 62H12.