Functional limit laws for depth quantiles

• Institute of Mathematical Finance, Ulm University, Ulm, Germany [giacomo.francisci@uni-ulm.de]

Keywords: Confidence regions – Donsker theorems – Multivariate quantiles – Statistical depth functions

Abstract

Statistical depth functions are bounded and non-negative functions $D(\cdot ,\cdot )$ from ${ℝ}^d$ and the space of probability distributions on ${ℝ}^d$, which are

• affine invariant,

• maximized at the center of symmetry for symmetric distributions,

• non-decreasing along any ray from a point of maximum, and

• vanishing at infinity

(see Liu [1990], Zuo and Serfling [2000]). Another useful property is upper semicontinuity, which ensures that the point of maximum in (iii) is attained. Specifically, for all probability distributions $P$ and $\alpha >0$, the upper-level sets

$$R_P(\alpha )=\{x\in {ℝ}^d:D(x,P)\ge \alpha \}$$

are compact [Dyckerhoff, 2004]. Multivariate quantile sets are defined by

$$Q_P(\alpha )=\partial R_P(\alpha ).$$

Assume without loss of generality that $D(\cdot ,P)$ is maximized at the origin. The set $Q_P(\alpha )$ may be identified with the radius function

$$r_P:S^{d-1}\to [0,\mathrm{\infty })$$

given for all directions $u$ in the unit sphere $S^{d-1}$ by

$$r_P(u)=\max \{s\ge 0:D(su,P)\ge \alpha \}.$$

Let $P_n$ be the empirical measure. We show that, under suitable assumptions,

$$\sqrt{n}(r_{P_n}-r_P)$$

converges in the space of bounded functions on $S^{d-1}$ to a Gaussian process and express the covariance function in terms of $D(\cdot ,P)$. Applications of our results include confidence regions and hypothesis testing for multivariate quantiles. Related results for halfspace depth are given by Nolan [1992].