2024/09/06 by Jordan Stoyanov, Wei, Chengfu, Stoyanov, Jordan +4
Engineering · Mathematics · #60G42 #62F25 #62L12 #Advanced Statistical Methods and Models #Control Systems and Identification #FOS: Mathematics #Probability (math.PR) #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2409.04198
openalex publication_date 2024/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 α∈ (0, 1], we assume that there is a known upper bound να > 0 for the (1 + α)-th central moment of the population distribution that the sample follows. Our findings replicate and `optimize' results in the above references for α= 1 (i.e., in models with finite variance) and enhance the results for α< 1. Furthermore, by employing the stitching method, we derive an upper bound on the width of the CS as O (((log log t)/t)^\fracα1+α) for the shrinking rate as t increases, and O((log (1/δ))(α)/(1+α)) for the growth rate as δ 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 α-Catoni-type CS with the bound in the above cited paper indicates that our CS achieves tighter width.