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We present canonical quantiles and depths for directional data following adistribution which is elliptically symmetric about a direction $\mu$ on thesphere $\mathcal{S}^{d-1}$. Our approach extends the concept of Ley et al. [1],which provides valuable geometric properties of the depth contours (such asconvexity and rotational equivariance) and a Bahadur-type representation of thequantiles. Their concept is canonical for rotationally symmetric depthcontours. However, it also produces rotationally symmetric depth contours whenthe underlying distribution is not rotationally symmetric. We solve this lackof flexibility for distributions with elliptical depth contours. The basic ideais to deform the elliptic contours by a diffeomorphic mapping to rotationallysymmetric contours, thus reverting to the canonical case in Ley et al. [1]. AMonte Carlo simulation study confirms our results. We use our method toevaluate the ellipticity of depth contours and for trimming of directionaldata. The analysis of fibre directions in fibre-reinforced concrete underlinesthe practical relevance.

Quantiles and depth for directional data from elliptically symmetric distributions