## Annals of Statistics

### Second-order correctness of the Poisson bootstrap

#### 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 $X = 0$.

#### Article information

Source
Ann. Statist., Volume 27, Number 5 (1999), 1666-1683.

Dates
First available in Project Euclid: 23 September 2004

https://projecteuclid.org/euclid.aos/1017939146

Digital Object Identifier
doi:10.1214/aos/1017939146

Mathematical Reviews number (MathSciNet)
MR2001c:62059

Zentralblatt MATH identifier
0965.62037

Subjects
Primary: 62G09: Resampling methods
Secondary: 60F05: Central limit and other weak theorems

#### Citation

Babu, G. Jogesh; Pathak, P. K.; Rao, C. R. Second-order correctness of the Poisson bootstrap. Ann. Statist. 27 (1999), no. 5, 1666--1683. doi:10.1214/aos/1017939146. https://projecteuclid.org/euclid.aos/1017939146

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• PENNSYLVANIA STATE UNIVERSITY EAST LANSING, MICHIGAN
• UNIVERSITY PARK, PENNSYLVANIA 16802 E-MAIL: babu@stat.psu.edu