Improved Catoni-Type Confidence Sequences for Estimating the Mean When the Variance Is Infinite

TL;DR

This paper refines Catoni-type confidence sequences for mean estimation in models with infinite variance, achieving tighter bounds and improved shrinking rates by leveraging known moment bounds and the stitching method.

Contribution

It introduces improved confidence sequences for infinite variance models, optimizing bounds and shrinking rates, extending previous results for finite variance cases.

Findings

Derived upper bounds on confidence sequence width with logarithmic factors

Achieved tighter confidence bounds compared to previous methods

Validated improvements through simulation experiments

Abstract

We consider a discrete time stochastic model with infinite variance and study the mean estimation problem as in Wang and Ramdas (2023). We refine the Catoni-type confidence sequence (abbr. CS) and use an idea of Bhatt et al. (2022) to achieve notable improvements of some currently existing results for such model. Specifically, for given $\alpha \in (0, 1]$, we assume that there is a known upper bound $\nu_{\alpha} > 0$ for the $(1 + \alpha)$-th central moment of the population distribution that the sample follows. Our findings replicate and `optimize' results in the above references for $\alpha = 1$ (i.e., in models with finite variance) and enhance the results for $\alpha < 1$. Furthermore, by employing the stitching method, we derive an upper bound on the width of the CS as $\mathcal{O} \left(((\log \log t)/t)^{\frac{\alpha}{1+\alpha}}\right)$ for the shrinking rate as $t$ increases, and $\mathcal{O}(\left(\log (1/\delta)\right)^{\frac{\alpha }{1+\alpha}})$ for the growth rate as $\delta$ decreases. These bounds are improving upon the bounds found in Wang and Ramdas (2023). Our theoretical results are illustrated by results from a series of simulation experiments. Comparing the performance of our improved $\alpha$-Catoni-type CS with the bound in the above cited paper indicates that our CS achieves tighter width.