Second-order properties of regeneration-based bootstrap for atomic Markov chains

In this paper, we show how the original bootstrap method introduced by Datta and McCormick (Can J Stat 21(2):181-193, 1993b), namely the regeneration-based bootstrap, for approximating the sampling distribution of sample mean statistics in the atomic Markovian setting can be modified to get the second-order accuracy. We prove that the drawback of the original construction mainly relies on an inaccurate estimation of the skewness of the sampling distribution by the bootstrap distribution and that it is possible to correct it by standardizing the regeneration-based bootstrap statistic by the length of the bootstrap series instead of the length of the sample and recentering the bootstrap distribution. An asymptotic result establishing the second-order accuracy of this bootstrap estimate up to O(n^-1log (n)) (close to the rate obtained in the i.i.d. setting) is also stated under weak moment assumptions.