### Abstract

The sampling distribution of several commonly occurring statistics are known to be closer to the corresponding bootstrap distribution than the normal distribution, under some conditions on the moments and the smoothness of the population distribution. These conditional approximations are suggestive of the unconditional ones considered in this paper, though one cannot be derived from the other by elementary methods. In this paper, probabilistic bounds are provided for the deviation of the sampling distribution from the bootstrap distribution. The rate of convergence to one, of the probability that the bootstrap approximation outperforms the normal approximation, is obtained. These rates can be applied to obtain the L^{p} bounds of Bhattacharya and Qumsiyeh (1989) under weaker conditions. The results apply to studentized versions of functions of multivariate means and thus cover a wide class of common statistics. As a consequence we also obtain approximations to percentiles of studentized means and their appropriate modifications. The results indicate the accuracy of the bootstrap confidence intervals both in terms of the actual coverage probability achieved and also the limits of the confidence interval.

Original language | English (US) |
---|---|

Pages (from-to) | 301-316 |

Number of pages | 16 |

Journal | Probability Theory and Related Fields |

Volume | 90 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1 1991 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Analysis
- Mathematics(all)

### Cite this

*Probability Theory and Related Fields*,

*90*(3), 301-316. https://doi.org/10.1007/BF01193748

}

*Probability Theory and Related Fields*, vol. 90, no. 3, pp. 301-316. https://doi.org/10.1007/BF01193748

**Accuracy of the bootstrap approximation.** / Bose, Arup; Babu, G. Jogesh.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Accuracy of the bootstrap approximation

AU - Bose, Arup

AU - Babu, G. Jogesh

PY - 1991/9/1

Y1 - 1991/9/1

N2 - The sampling distribution of several commonly occurring statistics are known to be closer to the corresponding bootstrap distribution than the normal distribution, under some conditions on the moments and the smoothness of the population distribution. These conditional approximations are suggestive of the unconditional ones considered in this paper, though one cannot be derived from the other by elementary methods. In this paper, probabilistic bounds are provided for the deviation of the sampling distribution from the bootstrap distribution. The rate of convergence to one, of the probability that the bootstrap approximation outperforms the normal approximation, is obtained. These rates can be applied to obtain the Lp bounds of Bhattacharya and Qumsiyeh (1989) under weaker conditions. The results apply to studentized versions of functions of multivariate means and thus cover a wide class of common statistics. As a consequence we also obtain approximations to percentiles of studentized means and their appropriate modifications. The results indicate the accuracy of the bootstrap confidence intervals both in terms of the actual coverage probability achieved and also the limits of the confidence interval.

AB - The sampling distribution of several commonly occurring statistics are known to be closer to the corresponding bootstrap distribution than the normal distribution, under some conditions on the moments and the smoothness of the population distribution. These conditional approximations are suggestive of the unconditional ones considered in this paper, though one cannot be derived from the other by elementary methods. In this paper, probabilistic bounds are provided for the deviation of the sampling distribution from the bootstrap distribution. The rate of convergence to one, of the probability that the bootstrap approximation outperforms the normal approximation, is obtained. These rates can be applied to obtain the Lp bounds of Bhattacharya and Qumsiyeh (1989) under weaker conditions. The results apply to studentized versions of functions of multivariate means and thus cover a wide class of common statistics. As a consequence we also obtain approximations to percentiles of studentized means and their appropriate modifications. The results indicate the accuracy of the bootstrap confidence intervals both in terms of the actual coverage probability achieved and also the limits of the confidence interval.

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UR - http://www.scopus.com/inward/citedby.url?scp=0010892124&partnerID=8YFLogxK

U2 - 10.1007/BF01193748

DO - 10.1007/BF01193748

M3 - Article

VL - 90

SP - 301

EP - 316

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 3

ER -