Empirical likelihood for Fr\'echet means on open books

TL;DR

This paper develops an empirical likelihood method for the Fréchet mean on open book spaces, a type of stratified metric space, providing theoretical results and bootstrap calibration for improved confidence region accuracy.

Contribution

It introduces an EL approach for Fréchet means on open book spaces, extending non-Euclidean EL theory to non-manifold stratified spaces with theoretical guarantees.

Findings

Derives a version of Wilks' theorem for the EL statistic.

Shows bootstrap calibration achieves $O(n^{-2})$ coverage error.

Connects the geometry of open books to the behavior of Fréchet means.

Abstract

Empirical Likelihood (EL) is a type of nonparametric likelihood that is useful in many statistical inference problems, including confidence region construction and $k$-sample problems. It enjoys some remarkable theoretical properties, notably Bartlett correctability. One area where EL has potential but is under-developed is in non-Euclidean statistics where the Fr\'echet mean is the population characteristic of interest. Only recently has a general EL method been proposed for smooth manifolds. In this work, we continue progress in this direction and develop an EL method for the Fr\'echet mean on a stratified metric space that is not a manifold: the open book, obtained by gluing copies of a Euclidean space along their common boundaries. The structure of an open book captures the essential behaviour of the Fr\'echet mean around certain singular regions of more general stratified spaces for complex data objects, and relates intimately to the local geometry of non-binary trees in the well-studied phylogenetic treespace. We derive a version of Wilks' theorem for the EL statistic, and elucidate on the delicate interplay between the asymptotic distribution and topology of the neighbourhood around the population Fr\'echet mean. We then present a bootstrap calibration of the EL, which proves that under mild conditions, bootstrap calibration of EL confidence regions have coverage error of size $O(n^{-2})$ rather than $O(n^{-1})$.