### Abstract

Estimation of confidence limits and intervals for the two- and three-parameter Weibull distributions are presented based on the methods of moment (MOM), probability weighted moments (PWM), and maximum likelihood (ML). The asymptotic variances of the MOM, PWM, and ML quantile estimators are derived as a function of the sample size, return period, and parameters. Such variances can be used for estimating the confidence limits and confidence intervals of the population quantiles. Except for the two-parameter Weibull model, the formulas obtained do not have simple forms but can be evaluated numerically. Simulation experiments were performed to verify the applicability of the derived confidence intervals of quantiles. The results show that overall, the ML method for estimating the confidence limits performs better than the other two methods in terms of bias and mean square error. This is specially so for λ > 0.5 even for small sample sizes (e.g. N = 10). However, the drawback of the ML method for determining the confidence limits is that it requires that the shape parameter be bigger than 2. The Weibull model based on the MOM, ML, and PWM estimation methods was applied to fit the distribution of annual 7-day low flows and 6-h maximum annual rainfall data. The results showed that the differences in the estimated quantiles based on the three methods are not large, generally are less than 10%. However, the differences between the confidence limits and confidence intervals obtained by the three estimation methods may be more significant. For instance, for the 7-day low flows the ratio between the estimated confidence interval to the estimated quantile based on ML is about 17% for T ≥ 2 while it is about 30% for estimation based on MOM and PWM methods.

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

Pages (from-to) | 284-309 |

Number of pages | 26 |

Journal | Stochastic Environmental Research and Risk Assessment |

Volume | 15 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2001 Dec 1 |

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### All Science Journal Classification (ASJC) codes

- Environmental Engineering
- Environmental Chemistry
- Water Science and Technology
- Safety, Risk, Reliability and Quality
- Environmental Science(all)

### Cite this

*Stochastic Environmental Research and Risk Assessment*,

*15*(4), 284-309. https://doi.org/10.1007/s004770100071

}

*Stochastic Environmental Research and Risk Assessment*, vol. 15, no. 4, pp. 284-309. https://doi.org/10.1007/s004770100071

**Estimation of confidence intervals of quantiles for the Weibull distribution.** / Heo, Jun-Haeng; Salas, J. D.; Kim, K. D.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Estimation of confidence intervals of quantiles for the Weibull distribution

AU - Heo, Jun-Haeng

AU - Salas, J. D.

AU - Kim, K. D.

PY - 2001/12/1

Y1 - 2001/12/1

N2 - Estimation of confidence limits and intervals for the two- and three-parameter Weibull distributions are presented based on the methods of moment (MOM), probability weighted moments (PWM), and maximum likelihood (ML). The asymptotic variances of the MOM, PWM, and ML quantile estimators are derived as a function of the sample size, return period, and parameters. Such variances can be used for estimating the confidence limits and confidence intervals of the population quantiles. Except for the two-parameter Weibull model, the formulas obtained do not have simple forms but can be evaluated numerically. Simulation experiments were performed to verify the applicability of the derived confidence intervals of quantiles. The results show that overall, the ML method for estimating the confidence limits performs better than the other two methods in terms of bias and mean square error. This is specially so for λ > 0.5 even for small sample sizes (e.g. N = 10). However, the drawback of the ML method for determining the confidence limits is that it requires that the shape parameter be bigger than 2. The Weibull model based on the MOM, ML, and PWM estimation methods was applied to fit the distribution of annual 7-day low flows and 6-h maximum annual rainfall data. The results showed that the differences in the estimated quantiles based on the three methods are not large, generally are less than 10%. However, the differences between the confidence limits and confidence intervals obtained by the three estimation methods may be more significant. For instance, for the 7-day low flows the ratio between the estimated confidence interval to the estimated quantile based on ML is about 17% for T ≥ 2 while it is about 30% for estimation based on MOM and PWM methods.

AB - Estimation of confidence limits and intervals for the two- and three-parameter Weibull distributions are presented based on the methods of moment (MOM), probability weighted moments (PWM), and maximum likelihood (ML). The asymptotic variances of the MOM, PWM, and ML quantile estimators are derived as a function of the sample size, return period, and parameters. Such variances can be used for estimating the confidence limits and confidence intervals of the population quantiles. Except for the two-parameter Weibull model, the formulas obtained do not have simple forms but can be evaluated numerically. Simulation experiments were performed to verify the applicability of the derived confidence intervals of quantiles. The results show that overall, the ML method for estimating the confidence limits performs better than the other two methods in terms of bias and mean square error. This is specially so for λ > 0.5 even for small sample sizes (e.g. N = 10). However, the drawback of the ML method for determining the confidence limits is that it requires that the shape parameter be bigger than 2. The Weibull model based on the MOM, ML, and PWM estimation methods was applied to fit the distribution of annual 7-day low flows and 6-h maximum annual rainfall data. The results showed that the differences in the estimated quantiles based on the three methods are not large, generally are less than 10%. However, the differences between the confidence limits and confidence intervals obtained by the three estimation methods may be more significant. For instance, for the 7-day low flows the ratio between the estimated confidence interval to the estimated quantile based on ML is about 17% for T ≥ 2 while it is about 30% for estimation based on MOM and PWM methods.

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U2 - 10.1007/s004770100071

DO - 10.1007/s004770100071

M3 - Article

VL - 15

SP - 284

EP - 309

JO - Stochastic Environmental Research and Risk Assessment

JF - Stochastic Environmental Research and Risk Assessment

SN - 1436-3240

IS - 4

ER -