Testing for practically significant dependencies in high dimensions via bootstrapping maxima of U-statistics
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866916122716012544 |
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| author | Bastian, Patrick Dette, Holger Heiny, Johannes |
| author_facet | Bastian, Patrick Dette, Holger Heiny, Johannes |
| contents | This paper takes a different look on the problem of testing the mutual independence of the components of a high-dimensional vector. Instead of testing if all pairwise associations (e.g. all pairwise Kendall's $τ$) between the components vanish, we are interested in the (null)-hypothesis that all pairwise associations do not exceed a certain threshold in absolute value. The consideration of these hypotheses is motivated by the observation that in the high-dimensional regime, it is rare, and perhaps impossible, to have a null hypothesis that can be exactly modeled by assuming that all pairwise associations are precisely equal to zero.
The formulation of the null hypothesis as a composite hypothesis makes the problem of constructing tests non-standard and in this paper we provide a solution for a broad class of dependence measures, which can be estimated by $U$-statistics. In particular we develop an asymptotic and a bootstrap level $α$-test for the new hypotheses in the high-dimensional regime. We also prove that the new tests are minimax-optimal and investigate their finite sample properties by means of a small simulation study and a data example. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_17439 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Testing for practically significant dependencies in high dimensions via bootstrapping maxima of U-statistics Bastian, Patrick Dette, Holger Heiny, Johannes Statistics Theory Probability 62G10, 62F40, 62C20 This paper takes a different look on the problem of testing the mutual independence of the components of a high-dimensional vector. Instead of testing if all pairwise associations (e.g. all pairwise Kendall's $τ$) between the components vanish, we are interested in the (null)-hypothesis that all pairwise associations do not exceed a certain threshold in absolute value. The consideration of these hypotheses is motivated by the observation that in the high-dimensional regime, it is rare, and perhaps impossible, to have a null hypothesis that can be exactly modeled by assuming that all pairwise associations are precisely equal to zero. The formulation of the null hypothesis as a composite hypothesis makes the problem of constructing tests non-standard and in this paper we provide a solution for a broad class of dependence measures, which can be estimated by $U$-statistics. In particular we develop an asymptotic and a bootstrap level $α$-test for the new hypotheses in the high-dimensional regime. We also prove that the new tests are minimax-optimal and investigate their finite sample properties by means of a small simulation study and a data example. |
| title | Testing for practically significant dependencies in high dimensions via bootstrapping maxima of U-statistics |
| topic | Statistics Theory Probability 62G10, 62F40, 62C20 |
| url | https://arxiv.org/abs/2210.17439 |