Comment Mia Hubert ∗ , Peter J. Rousseeuw † and Karlien Vanden Branden ‡ May 20, 2004 1 General comments We congratulate the authors (henceforth [MM]) on their highly original and thought-provoking paper. We will try to provide some additional insight and attempt to shed more light on a few of the open problems. The Student depth of some (µ, σ) w.r.t. a probability measure P on R is given by d(µ, σ, P) = inf (u 1 ,u 2 )=0 P  y : u 1 (y − µ)+ u 2 ((y − µ) 2 − σ 2 ) ≥ 0  . (1) The Student median is then the deepest fit, that is T (P) = (µ S ,σ S ) T = argmax µ,σ d(µ, σ, P). (2) It is instructive to consider (1) for a given couple (µ, σ) from the viewpoint of the two- dimensional (transformed) observations (y − µ, (y − µ) 2 − σ 2 ) which lie on a parabola. The depth of (µ, σ) is the smallest proportion of observations (or probability mass) lying in any halfspace HS with boundary line passing through the origin. To illustrate this we generated 20 data points from the standard normal distribution. In Figure 1(a) we see the parabola based on the inaccurate estimate (µ, σ) = (1, √ 2). The depth of this fit is zero, as illustrated by the halfspace HS. If instead we use the Student median (µ S ,σ S )=(−0.14, 0.39) in Figure 1(b), the depth is increased to 8/20 which is the maximal value for these data. * Assistant Professor, Department of Mathematics, Katholieke Universiteit Leuven, W. De Croylaan 54, B-3001 Leuven, Belgium, mia.hubert@wis.kuleuven.ac.be † Professor, Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1