Catoni-style Confidence Sequences under Infinite Variance
This work addresses a foundational statistical challenge for researchers and practitioners needing reliable confidence intervals in heavy-tailed or infinite-variance data scenarios, representing a strong specific gain rather than an incremental improvement.
The paper tackles the problem of constructing confidence sequences for data with infinite or non-existent variance, deriving tight Catoni-style sequences for distributions with bounded p-th moments (p in (1,2]) and improving upon existing finite variance results, showing better performance than Dubins-Savage inequality-based sequences.
In this paper, we provide an extension of confidence sequences for settings where the variance of the data-generating distribution does not exist or is infinite. Confidence sequences furnish confidence intervals that are valid at arbitrary data-dependent stopping times, naturally having a wide range of applications. We first establish a lower bound for the width of the Catoni-style confidence sequences for the finite variance case to highlight the looseness of the existing results. Next, we derive tight Catoni-style confidence sequences for data distributions having a relaxed bounded~$p^{th}-$moment, where~$p \in (1,2]$, and strengthen the results for the finite variance case of~$p =2$. The derived results are shown to better than confidence sequences obtained using Dubins-Savage inequality.