### Abstract

Significant reduction in the value of the number of bootstrap resamples may be achieved when exponential tilting is used to estimate a tail probability or a quantile, associated with a small probability, say p. However, exponential tilting is based on the assumption that the bootstrap distribution is approximately normal. In this paper, chisquare tilting is introduced when the bootstrap distribution of the statistics of interest is assumed to have a skewed distribution. In the case of small value of the new tilting reduces the asymptotic variance by at least a factor of 6, implies a large saving in bootstrap resamples. Calculations show that the variance reduction is very close to the asymptotic one even for sample sizes as small as 15. Some asymptotic results are given. Computer simulation as well as real data is used to illustrate the computation for the lower bootstrap confidence limit of the bootstrap sample variance. Although, bootstrap confidence intervals for a variance are known to perform very badly in practice even when sampling from a normal distribution, see Schenker (1987), our simulation indicates that our method perform reasonably well.

Original language | English |
---|---|

Pages (from-to) | 181-198 |

Number of pages | 18 |

Journal | Metron |

Volume | 60 |

Issue number | 3-4 |

Publication status | Published - 2002 |

### Fingerprint

### Keywords

- Bootstrap
- Chi-square Tilting
- Confidence limits
- Exponential Tilting
- Importance resample
- Sample variance

### ASJC Scopus subject areas

- Statistics and Probability

### Cite this

*Metron*,

*60*(3-4), 181-198.

**Importance resampling using Chi-Square Tilting.** / Samawi, Hani M.; Mustafa, Ahmad S Bany; Ahmed, Mohammad S.

Research output: Contribution to journal › Article

*Metron*, vol. 60, no. 3-4, pp. 181-198.

}

TY - JOUR

T1 - Importance resampling using Chi-Square Tilting

AU - Samawi, Hani M.

AU - Mustafa, Ahmad S Bany

AU - Ahmed, Mohammad S.

PY - 2002

Y1 - 2002

N2 - Significant reduction in the value of the number of bootstrap resamples may be achieved when exponential tilting is used to estimate a tail probability or a quantile, associated with a small probability, say p. However, exponential tilting is based on the assumption that the bootstrap distribution is approximately normal. In this paper, chisquare tilting is introduced when the bootstrap distribution of the statistics of interest is assumed to have a skewed distribution. In the case of small value of the new tilting reduces the asymptotic variance by at least a factor of 6, implies a large saving in bootstrap resamples. Calculations show that the variance reduction is very close to the asymptotic one even for sample sizes as small as 15. Some asymptotic results are given. Computer simulation as well as real data is used to illustrate the computation for the lower bootstrap confidence limit of the bootstrap sample variance. Although, bootstrap confidence intervals for a variance are known to perform very badly in practice even when sampling from a normal distribution, see Schenker (1987), our simulation indicates that our method perform reasonably well.

AB - Significant reduction in the value of the number of bootstrap resamples may be achieved when exponential tilting is used to estimate a tail probability or a quantile, associated with a small probability, say p. However, exponential tilting is based on the assumption that the bootstrap distribution is approximately normal. In this paper, chisquare tilting is introduced when the bootstrap distribution of the statistics of interest is assumed to have a skewed distribution. In the case of small value of the new tilting reduces the asymptotic variance by at least a factor of 6, implies a large saving in bootstrap resamples. Calculations show that the variance reduction is very close to the asymptotic one even for sample sizes as small as 15. Some asymptotic results are given. Computer simulation as well as real data is used to illustrate the computation for the lower bootstrap confidence limit of the bootstrap sample variance. Although, bootstrap confidence intervals for a variance are known to perform very badly in practice even when sampling from a normal distribution, see Schenker (1987), our simulation indicates that our method perform reasonably well.

KW - Bootstrap

KW - Chi-square Tilting

KW - Confidence limits

KW - Exponential Tilting

KW - Importance resample

KW - Sample variance

UR - http://www.scopus.com/inward/record.url?scp=84891468601&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84891468601&partnerID=8YFLogxK

M3 - Article

VL - 60

SP - 181

EP - 198

JO - Metron

JF - Metron

SN - 0026-1424

IS - 3-4

ER -