Second-order correctness of the Poisson bootstrap

TLDR

Sequential bootstrap resamples until about 63 % of the original observations appear, yet proving its second‑order correctness remains challenging. This study investigates the second‑order correctness of the sequential bootstrap. We approximate the sequential scheme with a Poisson(1) resampling process censored at zero. Previously we showed that the sequential bootstrap’s empirical characteristics differ from the ordinary bootstrap by at most O(n⁻³⁄⁴).

Abstract

Rao, Pathak and Koltchinskii have recently studied a sequential approach to resampling in which resampling is carried out sequentially one-by-one (with replacement each time) until the bootstrap sample contains $m \approx (1 - e^{-1})n \approx 0.632n$ distinct observations from the original sample. In our previous work, we have established that the main empirical characteristics of the sequential bootstrap go through, in the sense of being within a distance $O(n^{-3/4})$ from those of the usual bootstrap. However, the theoretical justification of the second-order correctness of the sequential bootstrap is somewhat difficult. It is the main topic of this investigation. Among other things, we accomplish it by approximating our sequential scheme by a resampling scheme based on the Poisson distribution with mean $\mu = 1$ and censored at $X = 0$.