Asymptotic normality and confidence region for Catoni's Z estimator

Abstract

This paper studies the asymptotic property and confidence region of Catoni's Z estimator under the finite variance assumption. First, we investigate the CLT for Catoni's Z estimator with the asymptotically optimal variance, if the tuning parameter has order . Second, we propose the normal approximated confidence region of Catoni's Z estimator. The simulations show that the proposed 99% CLT‐based confidence regions are shorter than the existing non‐asymptotic confidence regions; mean squared errors of Catoni's Z estimators are typically smaller than sample means from heavy‐tailed data. Furthermore, the simulated coverage probabilities of the CLT‐based confidence region are close to the confidence levels when the sample size is moderately large. For 6‐year DOCVIS data (the number of doctor visits) in the German health care demand dataset, our CLT‐based confidence regions are tighter than non‐asymptotic confidence regions.