Empirical likelihood for Fréchet means on open books
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arXiv
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| Format: | Preprint |
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2024
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| author | Bharath, Karthik Le, Huiling Wood, Andrew T A Yan, Xi |
| author_facet | Bharath, Karthik Le, Huiling Wood, Andrew T A Yan, Xi |
| contents | 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 Fréchet 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 Fréchet 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 Fréchet 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 Fréchet 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 $O(n^{-1})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18818 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Empirical likelihood for Fréchet means on open books Bharath, Karthik Le, Huiling Wood, Andrew T A Yan, Xi Statistics Theory Computation Methodology 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 Fréchet 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 Fréchet 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 Fréchet 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 Fréchet 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 $O(n^{-1})$. |
| title | Empirical likelihood for Fréchet means on open books |
| topic | Statistics Theory Computation Methodology |
| url | https://arxiv.org/abs/2412.18818 |