Catoni-style confidence sequences for heavy-tailed mean estimation

Hongjian Wang and Aaditya Ramdas

Stochastic Processes and their Applications, 2023, vol. 163, issue C, 168-202

Abstract: A confidence sequence (CS) is a sequence of confidence intervals that is valid at arbitrary data-dependent stopping times. These are useful in applications like A/B testing, multi-armed bandits, off-policy evaluation, election auditing, etc. We present three approaches to constructing a confidence sequence for the population mean, under the minimal assumption that only an upper bound σ2 on the variance is known. While previous works rely on light-tail assumptions like boundedness or subGaussianity (under which all moments of a distribution exist), the confidence sequences in our work are able to handle data from a wide range of heavy-tailed distributions. The best among our three methods — the Catoni-style confidence sequence — performs remarkably well in practice, essentially matching the state-of-the-art methods for σ2-subGaussian data, and provably attains the loglogt/t lower bound due to the law of the iterated logarithm. Our findings have important implications for sequential experimentation with unbounded observations, since the σ2-bounded-variance assumption is more realistic and easier to verify than σ2-subGaussianity (which implies the former). We also extend our methods to data with infinite variance, but having pth central moment (1Keywords: Confidence sequences; Supermartingales; Sequential analysis; Law of the iterated logarithm; Heavy-tailed distributions (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1016/j.spa.2023.05.007

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