Testing Imprecise Hypotheses
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912850983780352 |
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| author | Kania, Lucas Manole, Tudor Wasserman, Larry Balakrishnan, Sivaraman |
| author_facet | Kania, Lucas Manole, Tudor Wasserman, Larry Balakrishnan, Sivaraman |
| contents | Many scientific applications involve testing theories that are only partially specified. This task often amounts to testing the goodness-of-fit of a candidate distribution while allowing for reasonable deviations from it. The tolerant testing framework provides a systematic way of constructing such tests. Rather than testing the simple null hypothesis that data was drawn from a candidate distribution, a tolerant test assesses whether the data is consistent with any distribution that lies within a given neighborhood of the candidate. As this neighborhood grows, the tolerance to misspecification increases, while the power of the test decreases. In this work, we characterize the information-theoretic trade-off between the size of the neighborhood and the power of the test, in several canonical models. On the one hand, we characterize the optimal trade-off for tolerant testing in the Gaussian sequence model, under deviations measured in both smooth and non-smooth norms. On the other hand, we study nonparametric analogues of this problem in smooth regression and density models. Along the way, we establish the sub-optimality of the classical chi-squared statistic for tolerant testing, and study simple alternative hypothesis tests. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_20717 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Testing Imprecise Hypotheses Kania, Lucas Manole, Tudor Wasserman, Larry Balakrishnan, Sivaraman Statistics Theory Methodology Many scientific applications involve testing theories that are only partially specified. This task often amounts to testing the goodness-of-fit of a candidate distribution while allowing for reasonable deviations from it. The tolerant testing framework provides a systematic way of constructing such tests. Rather than testing the simple null hypothesis that data was drawn from a candidate distribution, a tolerant test assesses whether the data is consistent with any distribution that lies within a given neighborhood of the candidate. As this neighborhood grows, the tolerance to misspecification increases, while the power of the test decreases. In this work, we characterize the information-theoretic trade-off between the size of the neighborhood and the power of the test, in several canonical models. On the one hand, we characterize the optimal trade-off for tolerant testing in the Gaussian sequence model, under deviations measured in both smooth and non-smooth norms. On the other hand, we study nonparametric analogues of this problem in smooth regression and density models. Along the way, we establish the sub-optimality of the classical chi-squared statistic for tolerant testing, and study simple alternative hypothesis tests. |
| title | Testing Imprecise Hypotheses |
| topic | Statistics Theory Methodology |
| url | https://arxiv.org/abs/2510.20717 |