Quantiles and depth for directional data from elliptically and skew-rotationally symmetric distributions

Abstract

We present canonical quantiles and depths for directional data following a distribution which is elliptically symmetric about a direction μ on the sphere $S^{d-1}$. Our approach extends the concept of Ley et al. [1], which provides valuable geometric properties of the depth contours (such as convexity and rotational equivariance) and a Bahadur-type representation of the quantiles. Their concept is canonical for rotationally symmetric depth contours. However, it also produces rotationally symmetric depth contours when the underlying distribution is not rotationally symmetric. We solve this lack of flexibility for distributions with elliptical depth contours. The basic idea is to deform the elliptic contours by a diffeomorphic mapping to rotationally symmetric contours, thus reverting to the canonical case in Ley et al. [1]. Furthermore, an extension to skew-rotationally symmetric distributions is presented. A Monte Carlo simulation study confirms our results. We use our method to evaluate the ellipticity of depth contours and for trimming of directional data. The analysis of fibre directions in fibre-reinforced concrete underlines the practical relevance.