Uniform Confidence Band for Optimal Transport Map on One-Dimensional Data
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913234696536064 |
|---|---|
| author | Ponnoprat, Donlapark Okano, Ryo Imaizumi, Masaaki |
| author_facet | Ponnoprat, Donlapark Okano, Ryo Imaizumi, Masaaki |
| contents | We develop a statistical inference method for an optimal transport map between distributions on real numbers with uniform confidence bands. The concept of optimal transport (OT) is used to measure distances between distributions, and OT maps are used to construct the distance. OT has been applied in many fields in recent years, and its statistical properties have attracted much interest. In particular, since the OT map is a function, a uniform norm-based statistical inference is significant for visualization and interpretation. In this study, we derive a limit distribution of a uniform norm of an estimation error for the OT map, and then develop a uniform confidence band based on it. In addition to our limit theorem, we develop a bootstrap method with kernel smoothing, then also derive its validation and guarantee on an asymptotic coverage probability of the confidence band. Our proof is based on the functional delta method and the representation of OT maps on the reals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_09257 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Uniform Confidence Band for Optimal Transport Map on One-Dimensional Data Ponnoprat, Donlapark Okano, Ryo Imaizumi, Masaaki Statistics Theory 62G15, 62G20 We develop a statistical inference method for an optimal transport map between distributions on real numbers with uniform confidence bands. The concept of optimal transport (OT) is used to measure distances between distributions, and OT maps are used to construct the distance. OT has been applied in many fields in recent years, and its statistical properties have attracted much interest. In particular, since the OT map is a function, a uniform norm-based statistical inference is significant for visualization and interpretation. In this study, we derive a limit distribution of a uniform norm of an estimation error for the OT map, and then develop a uniform confidence band based on it. In addition to our limit theorem, we develop a bootstrap method with kernel smoothing, then also derive its validation and guarantee on an asymptotic coverage probability of the confidence band. Our proof is based on the functional delta method and the representation of OT maps on the reals. |
| title | Uniform Confidence Band for Optimal Transport Map on One-Dimensional Data |
| topic | Statistics Theory 62G15, 62G20 |
| url | https://arxiv.org/abs/2307.09257 |