General notions of statistical depth function. Annals Stat., 28(2):461–482, 2000. A Appendix: Proofs for Properties of ψ(q) In this Appendix we present proofs of both theorems presented in Section 3. Proof. [of Theorem 1] Consider the point q for which th (0)

by Y Zuo, R Serfling
Venue:Ψ0(q), . . . ,Ψn−1(q)〉, whose elements satisfy Ψj(q) = ψn−1−j(q) for j = 0, 1, . . . , n − 1. We next prove that Ψj(q) = ψj(q), implying the stated symmetry of the ψ histogram. To see this, consider the probability that the query point q is interior to CH