seMCD: Sequentially implemented Monte Carlo depth computation with statistical guarantees
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866908441049563136 |
|---|---|
| author | Gnettner, Felix Kirch, Claudia Nieto-Reyes, Alicia |
| author_facet | Gnettner, Felix Kirch, Claudia Nieto-Reyes, Alicia |
| contents | Statistical depth functions provide center-outward orderings in spaces of dimension larger than one, where a natural ordering does not exist. The numerical evaluation of such depth functions can be computationally prohibitive, even for relatively low dimensions. We present a novel sequentially implemented Monte Carlo methodology for the computation of, theoretical and empirical, depth functions and related quantities (seMCD), that outputs an interval, a so-called seMCD-bucket, to which the quantity of interest belongs with a high probability prespecified by the user. For specific classes of depth functions, we adapt algorithms from sequential testing, providing finite-sample guarantees. For depth functions dependent on unknown distributions, we offer asymptotic guarantees using non-parametric statistical methods. In contrast to plain-vanilla Monte Carlo methodology the number of samples required in the algorithm is random but typically much smaller than standard choices suggested in the literature. The seMCD method can be applied to various depth functions, covering multivariate and functional spaces. We demonstrate the efficiency and reliability of our approach through empirical studies, highlighting its applicability in outlier or anomaly detection, classification, and depth region computation. In conclusion, the seMCD-algorithm can achieve accurate depth approximations with few Monte Carlo samples while maintaining rigorous statistical guarantees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_06227 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | seMCD: Sequentially implemented Monte Carlo depth computation with statistical guarantees Gnettner, Felix Kirch, Claudia Nieto-Reyes, Alicia Methodology Statistics Theory Computation Machine Learning 62L12 (Primary) 62L20, 62R07, 68T09 (Secondary) Statistical depth functions provide center-outward orderings in spaces of dimension larger than one, where a natural ordering does not exist. The numerical evaluation of such depth functions can be computationally prohibitive, even for relatively low dimensions. We present a novel sequentially implemented Monte Carlo methodology for the computation of, theoretical and empirical, depth functions and related quantities (seMCD), that outputs an interval, a so-called seMCD-bucket, to which the quantity of interest belongs with a high probability prespecified by the user. For specific classes of depth functions, we adapt algorithms from sequential testing, providing finite-sample guarantees. For depth functions dependent on unknown distributions, we offer asymptotic guarantees using non-parametric statistical methods. In contrast to plain-vanilla Monte Carlo methodology the number of samples required in the algorithm is random but typically much smaller than standard choices suggested in the literature. The seMCD method can be applied to various depth functions, covering multivariate and functional spaces. We demonstrate the efficiency and reliability of our approach through empirical studies, highlighting its applicability in outlier or anomaly detection, classification, and depth region computation. In conclusion, the seMCD-algorithm can achieve accurate depth approximations with few Monte Carlo samples while maintaining rigorous statistical guarantees. |
| title | seMCD: Sequentially implemented Monte Carlo depth computation with statistical guarantees |
| topic | Methodology Statistics Theory Computation Machine Learning 62L12 (Primary) 62L20, 62R07, 68T09 (Secondary) |
| url | https://arxiv.org/abs/2507.06227 |