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| Autori principali: | , , , , |
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| Natura: | Preprint |
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2023
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2303.00471 |
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| _version_ | 1866914632571027456 |
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| author | Hao, Yunda Grünwald, Peter Lardy, Tyron Long, Long Adams, Reuben |
| author_facet | Hao, Yunda Grünwald, Peter Lardy, Tyron Long, Long Adams, Reuben |
| contents | We develop and compare e-variables for testing whether $k$ samples of data are drawn from the same distribution, the alternative being that they come from different elements of an exponential family. We consider the GRO (growth-rate optimal) e-variables for (1) a `small' null inside the same exponential family, and (2) a `large' nonparametric null, as well as (3) an e-variable arrived at by conditioning on the sum of the sufficient statistics. (2) and (3) are efficiently computable, and extend ideas from Turner et al. [2021] and Wald [1947] respectively from Bernoulli to general exponential families. We provide theoretical and simulation-based comparisons of these e-variables in terms of their logarithmic growth rate, and find that for small effects all four e-variables behave surprisingly similarly; for the Gaussian location and Poisson families, e-variables (1) and (3) coincide; for Bernoulli, (1) and (2) coincide; but in general, whether (2) or (3) grows faster under the alternative is family-dependent. We furthermore discuss algorithms for numerically approximating (1). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_00471 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | E-values for k-Sample Tests With Exponential Families Hao, Yunda Grünwald, Peter Lardy, Tyron Long, Long Adams, Reuben Methodology Statistics Theory We develop and compare e-variables for testing whether $k$ samples of data are drawn from the same distribution, the alternative being that they come from different elements of an exponential family. We consider the GRO (growth-rate optimal) e-variables for (1) a `small' null inside the same exponential family, and (2) a `large' nonparametric null, as well as (3) an e-variable arrived at by conditioning on the sum of the sufficient statistics. (2) and (3) are efficiently computable, and extend ideas from Turner et al. [2021] and Wald [1947] respectively from Bernoulli to general exponential families. We provide theoretical and simulation-based comparisons of these e-variables in terms of their logarithmic growth rate, and find that for small effects all four e-variables behave surprisingly similarly; for the Gaussian location and Poisson families, e-variables (1) and (3) coincide; for Bernoulli, (1) and (2) coincide; but in general, whether (2) or (3) grows faster under the alternative is family-dependent. We furthermore discuss algorithms for numerically approximating (1). |
| title | E-values for k-Sample Tests With Exponential Families |
| topic | Methodology Statistics Theory |
| url | https://arxiv.org/abs/2303.00471 |