Tolerant Testing in the Gaussian Sequence Model

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Recently, there has been interest in testing hypotheses under misspecification. In tolerant testing, the practitioner states a simple null hypothesis and indicates how much deviation from it should be tolerated when testing it. In this work, we study the tolerant testing problem in the Gaussian sequence model. Specifically, given an observation from a high-dimensional Gaussian distribution, is the p-norm of its mean less than δ (null hypothesis) or greater than ε (alternative hypothesis)? When δ = 0, the problem reduces to simple hypothesis testing, while δ > 0 indicates how much imprecision in the null hypothesis should be tolerated. Via the minimax hypothesis testing framework, we characterise the smallest separation between null and alternative hypotheses such that it is possible to consistently distinguish them. Extending the results of Ingster 2001, we find that as δ is increased, the hardness of the problem interpolates between simple hypothesis testing and functional estimation. Furthermore, our results show a strong connection to tolerant testing with multinomial data (Canonne et al. 2022).

minimax

hypothesis testing

tolerant testing

imprecise hypothesis

Gaussian sequence model

misspecification 

IMS