Annals of Statistics

Second-order correctness of the Poisson bootstrap

G. Jogesh Babu, P. K. Pathak, and C. R. Rao

Full-text: Open access


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

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

First available in Project Euclid: 23 September 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Edgeworth expansions bootstrap breakdown point expansions for conditional distributions lattice distribution


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.

Export citation


  • BABU, G. J. and BAI, Z. D. 1996. Mixtures of global and local Edgeworth expansions and their applications. J. Multivariate Anal. 59 282 307. Z.
  • BABU, G. J. and SINGH, K. 1983. Inference on means using the bootstrap. Ann. Statist. 11 999 1003. Z.
  • BABU, G. J. and SINGH, K. 1984. On the term Edgeworth correction by Efron's bootstrap. Sankhya Ser. A 46 219 232. Z.
  • BABU, G. J. and SINGH, K. 1985. Edgeworth expansion for sampling without replacement from finite populations. J. Multivariate Anal. 17 261 278. Z.
  • BABU, G. J. and SINGH, K. 1989. On Edgeworth expansions in the mixture cases. Ann. Statist. 17 443 447. Z.
  • BAI, Z. D. and RAO, C. R. 1991. Edgeworth expansion of a function of sample means. Ann. Statist. 19 1295 1315. Z.
  • BAI, Z. D. and RAO, C. R. 1992. A note on the Edgeworth expansion for ratio of sample means. Sankhya Ser. A 54 309 322. Z.
  • BHATTACHARYA, R. N. and GHOSH, J. K. 1978. On the validity of the formal Edgeworth expansion. Ann. Statist. 6 434 451.Z.
  • BHATTACHARYA, R. N. and RANGA RAO, R. 1986. Normal Approximations and Asymptotic Expansions. Krieger, Malabar, FL. Z.
  • EFRON, B. 1979. Bootstrap methods: another look at the jackknife. Ann. Statist. 7 1 26.
  • GINE, E. and ZINN, J. 1990. Bootstrapping general empirical measures. Ann. Probab. 18 ´ 851 869. Z.
  • HALL, P. and MAMMEN, E. 1994. On general resampling algorithms and their performance in distribution estimation. Ann. Statist. 22 2011 2030. Z.
  • PATHAK, P. K. 1962. On simple random sampling with replacement. Sankhya Ser. A 24 287 302. Z.
  • PETROV, V. V. 1975. Sums of Independent Random Variables. Springer, New York. Z.
  • RAO, C. R., PATHAK, P. K. and KOLTCHINSKII, V. I. 1997. Bootstrap by sequential resampling. J. Statist. Plann. Inference 64 257 281. Z.
  • SINGH, K. 1981. On the asymptotic accuracy of Efron's bootstrap. Ann. Statist. 9 1187 1195. Z.
  • SWEETING, T. J. 1977. Speeds of convergence for the multidimensional central limit theorem. Ann. Probab. 5 28 41.