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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2023, Vol. 40 ›› Issue (6): 1011-1020.doi: 10.3969/j.issn.1005-3085.2023.06.012

• • 上一篇    

广义Pareto分布参数的Bootstrap置信区间及应用

张艳芳1,2,  赵宜宾1,  任晴晴1   

  1. 1. 防灾科技学院，河北 三河 065201
    2. 河北省地震动力学重点实验室，河北 三河 065201
  • 收稿日期:2021-08-26 接受日期:2022-06-21 出版日期:2023-12-15 发布日期:2024-02-15
  • 通讯作者: 赵宜宾 E-mail: zhaoyibin5362@126.com
  • 基金资助:
    中央高校基本科研业务费专题项目 (ZY20215140)；河北省教育厅科学技术研究项目 (ZD2022160)；河,北省高等学校科学技术研究项目 (Z2020224).

Bootstrap Confidence Interval for Parameters of Generalized Pareto Distribution and Its Application

ZHANG Yanfang1,2,  ZHAO Yibin1,  REN Qingqing1   

  1. 1. Institute of Disaster Prevention, Sanhe, Hebei 065201
    2. Hebei Key Laboratory of Earthquake Dynamics, Sanhe, Hebei 065201
  • Received:2021-08-26 Accepted:2022-06-21 Online:2023-12-15 Published:2024-02-15
  • Contact: Y. Zhao. E-mail address: zhaoyibin5362@126.com
  • Supported by:
    The Fundamental Research Funds for the Central Universities (ZY20215140); the Science and Technology Project of Hebei Education Department (ZD2022160); the Science and Technology Research Project for Colleges and Universities in Hebei Province (Z2020224).

摘要：

针对广义Pareto分布应用于超阈值数据拟合时尾部数据较少的情况，将参数的极大似然估计与Bootstrap方法相结合，给出参数的Bootstrap置信区间。Bootstrap置信区间计算方法既可以通过重复抽样充分利用样本信息，又可以避免样本较少时估计量方差偏大的情况。理论证明参数的Bootstrap置信区间是渐近有效的。数值模拟结果表明在一定的置信度下，置信区间的区间长度和覆盖率比较合理。应用建立的Pareto模型对巴颜喀拉地震带震级数据进行分析，结果表明在相同的置信度下，采用Bootstrap方法计算的置信区间比由近似协方差矩阵计算的区间长度更短，这为广义Pareto模型描述地震震级分布的适用性提供了支撑。

关键词: 广义Pareto分布, Bootstrap置信区间, 震级分布, 参数估计

Abstract:

Since there are few tail data when the generalized Pareto distribution is applied to the data beyond the threshold, the Bootstrap confidence interval for the parameters is derived by combining the maximum likelihood estimation of the parameters with the Bootstrap method. The Bootstrap method can not only make full use of the sample information by resampling, but also avoid the estimation of the variance of the estimation when there are fewer samples. Bootstrap confidence intervals for parameters are proved being asymptotically efficient. Numerical simulation results show that the interval length and coverage are reasonable under a certain confidence level. The established Pareto model is used to analyze the magnitude data of the Bayan Har seismic zone. The results show that under the same confidence level, the length of the confidence interval determined by the Bootstrap method is shorter than that calculated by the approximate covariance matrix, which supports the applicability of the generalized Pareto model to describe the earthquake magnitude.

Key words: generalized Pareto distribution, Bootstrap confidence interval, magnitude distribution, parameter estimation

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