Catoni-style Confidence Sequences under Infinite Variance

TL;DR

This paper extends confidence sequences to distributions with infinite variance, providing tighter bounds and new methods for data with relaxed moment conditions, improving upon existing inequalities.

Contribution

It introduces tight Catoni-style confidence sequences for distributions with relaxed moment conditions, including infinite variance scenarios, advancing the theoretical understanding.

Findings

Derived confidence sequences outperform Dubins-Savage inequality-based sequences.

Established lower bounds highlighting looseness of previous results.

Extended confidence sequences to distributions with relaxed p-th moments.

Abstract

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.