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Catoni-style confidence sequences for heavy-tailed mean estimation

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  • Wang, Hongjian
  • Ramdas, Aaditya

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 (1

Suggested Citation

  • Wang, Hongjian & Ramdas, Aaditya, 2023. "Catoni-style confidence sequences for heavy-tailed mean estimation," Stochastic Processes and their Applications, Elsevier, vol. 163(C), pages 168-202.
    • Handle: RePEc:eee:spapps:v:163:y:2023:i:c:p:168-202
    DOI: 10.1016/j.spa.2023.05.007
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    References listed on IDEAS

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    1. Rasul A. Khan, 2009. "L p -Version of the Dubins–Savage Inequality and Some Exponential Inequalities," Journal of Theoretical Probability, Springer, vol. 22(2), pages 348-364, June.
    2. Csenki, Attila, 1979. "A note on confidence sequences in multiparameter exponential families," Journal of Multivariate Analysis, Elsevier, vol. 9(2), pages 337-340, June.
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