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

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.