E-Values for Exponential Families: the General Case
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912553820487680 |
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| author | Hao, Yunda Grünwald, Peter |
| author_facet | Hao, Yunda Grünwald, Peter |
| contents | We analyze common types of e-variables and e-processes for composite exponential family nulls: the optimal e-variable based on the reverse information projection (RIPr), the conditional (COND) e-variable, and the universal inference (UI) and sequen\-tialized RIPr e-processes. We characterize the RIPr prior for simple and Bayes-mixture based alternatives, either precisely (for Gaussian nulls and alternatives) or in an approximate sense (general exponential families). We provide conditions under which the RIPr e-variable is (again exactly vs. approximately) equal to the COND e-variable. Based on these and other interrelations which we establish, we determine the e-power of the four e-statistics as a function of sample size, exactly for Gaussian and up to $o(1)$ in general. For $d$-dimensional null and alternative, the e-power of UI tends to be smaller by a term of $(d/2) \log n + O(1)$ than that of the COND e-variable, which is the clear winner. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_11134 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | E-Values for Exponential Families: the General Case Hao, Yunda Grünwald, Peter Methodology We analyze common types of e-variables and e-processes for composite exponential family nulls: the optimal e-variable based on the reverse information projection (RIPr), the conditional (COND) e-variable, and the universal inference (UI) and sequen\-tialized RIPr e-processes. We characterize the RIPr prior for simple and Bayes-mixture based alternatives, either precisely (for Gaussian nulls and alternatives) or in an approximate sense (general exponential families). We provide conditions under which the RIPr e-variable is (again exactly vs. approximately) equal to the COND e-variable. Based on these and other interrelations which we establish, we determine the e-power of the four e-statistics as a function of sample size, exactly for Gaussian and up to $o(1)$ in general. For $d$-dimensional null and alternative, the e-power of UI tends to be smaller by a term of $(d/2) \log n + O(1)$ than that of the COND e-variable, which is the clear winner. |
| title | E-Values for Exponential Families: the General Case |
| topic | Methodology |
| url | https://arxiv.org/abs/2409.11134 |