Note on the spatial quantile of a random vector

Let [alpha] [set membership, variant] (0, 1); the [alpha]-quantile of an -valued (k [greater-or-equal, slanted] 1) random variable X is a point minimizing the expectation E(||X - [theta]||p,[alpha] - ||X||p,[alpha]), where ||·||p,[alpha] is defined in terms of the lp-norm, 1 [less-than-or-equals, slant] p [less-than-or-equals, slant] [infinity], and [alpha] [set membership, variant] (0, 1). The properties of such an [alpha]-quantile extend those obtained previously for [alpha] = 0.5, i.e. for the median (see Kemperman, 1987). Computational aspects are also discussed.