Centre-set theory for alternative U-statistic depths

Develop centre-set theory for metric spatial depth in Euclidean space and for metric lens depth on the real line, accounting for the vanishing of the influence function at the centre.

Background

The procedures in the paper can be applied to other U-statistic depths, including metric spatial depth and metric lens depth, and the multiplier band retains coverage under suitable kernel conditions. The detailed centre-set results in the paper, however, rely on features specific to the reference depth.

For metric spatial depth in Euclidean space and metric lens depth on the line, the influence function vanishes at the centre. Consequently, the estimation error has a different asymptotic order, and the corresponding centre-set results remain unresolved.

References

For metric spatial depth in Euclidean space, and for metric lens depth on the line, the influence function vanishes at the centre (Supplementary Section~S1), so the estimation error there has a different order, and centre sets for these depths remain open.

— Simultaneous Inference for Centre-Outward Rankings of Random Objects  (2610.18533 - Lyu et al., 13 Oct 2026) in Discussion, Section 7