Measuring Evidence against Exchangeability and Group Invariance with E-values
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866917261215793152 |
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| author | Koning, Nick W. |
| author_facet | Koning, Nick W. |
| contents | We study e-values for quantifying evidence against exchangeability and general invariance of a random variable under a compact group. We start by characterizing such e-values, and explaining how they nest traditional group invariance tests as a special case. We show they can be easily designed for an arbitrary test statistic, and computed through Monte Carlo sampling. We prove a result that characterizes optimal e-values for group invariance against optimality targets that satisfy a mild orbit-wise decomposition property. We apply this to design expected-utility-optimal e-values for group invariance, which include both Neyman-Pearson-optimal tests and log-optimal e-values. Moreover, we generalize the notion of rank- and sign-based testing to compact groups, by using a representative inversion kernel. In addition, we characterize e-processes for group invariance for arbitrary filtrations, and provide tools to construct them. We also describe test martingales under a natural filtration, which are simpler to construct. Peeking beyond compact groups, we encounter e-values and e-processes based on ergodic theorems. These nest e-processes based on de Finetti's theorem for testing exchangeability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_01153 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Measuring Evidence against Exchangeability and Group Invariance with E-values Koning, Nick W. Statistics Theory Methodology We study e-values for quantifying evidence against exchangeability and general invariance of a random variable under a compact group. We start by characterizing such e-values, and explaining how they nest traditional group invariance tests as a special case. We show they can be easily designed for an arbitrary test statistic, and computed through Monte Carlo sampling. We prove a result that characterizes optimal e-values for group invariance against optimality targets that satisfy a mild orbit-wise decomposition property. We apply this to design expected-utility-optimal e-values for group invariance, which include both Neyman-Pearson-optimal tests and log-optimal e-values. Moreover, we generalize the notion of rank- and sign-based testing to compact groups, by using a representative inversion kernel. In addition, we characterize e-processes for group invariance for arbitrary filtrations, and provide tools to construct them. We also describe test martingales under a natural filtration, which are simpler to construct. Peeking beyond compact groups, we encounter e-values and e-processes based on ergodic theorems. These nest e-processes based on de Finetti's theorem for testing exchangeability. |
| title | Measuring Evidence against Exchangeability and Group Invariance with E-values |
| topic | Statistics Theory Methodology |
| url | https://arxiv.org/abs/2310.01153 |