Statistical characteristics of earthquake magnitude based on the composite model

Yanfang Zhang; Fuchang Wang; Yibin Zhao; Yanfang Zhang; Fuchang Wang; Yibin Zhao

doi:10.3934/math.2024032

AIMS Mathematics

2024, Volume 9, Issue 1: 607-624. doi: 10.3934/math.2024032

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Theory article

Statistical characteristics of earthquake magnitude based on the composite model

Department of Mathematics, Institute of Disaster Prevention, No. 465, Xueyuan Street, Yanjiao High Tech Zone, Sanhe City, Hebei Province, 065201, China

Received: 11 August 2023 Revised: 11 November 2023 Accepted: 22 November 2023 Published: 01 December 2023
MSC : 62F10

Threshold selection is challenging when analyzing tail data with a generalized Pareto distribution. Data below the threshold was not used in the model, resulting in incomplete characterization of the whole data. This paper applied the Gamma distribution, Weibull distribution, and lognormal distribution to fit the central data separately, and a generalized Pareto distribution (GPD) was used to analyze the tail data. In such composite models, the thresholds are estimated directly as parameters. We proposed an empirical distribution function-based parameter estimation method. The absolute value of the difference between the empirical distribution function and the composite distribution function was used as a loss function to obtain an estimate of the parameter. This parameter estimation method is suitable for complex multiparameter distributions. The estimation method based on the empirical distribution function was verified to be feasible through simulation studies. The composite model and the estimation method based on the empirical distribution function were applied to study the earthquake magnitude data to provide a reference for earthquake hazard analysis.
- generalized Pareto distribution,
- composite model,
- empirical distribution function,
- seismic hazard
Citation: Yanfang Zhang, Fuchang Wang, Yibin Zhao. Statistical characteristics of earthquake magnitude based on the composite model[J]. AIMS Mathematics, 2024, 9(1): 607-624. doi: 10.3934/math.2024032

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Abstract

Threshold selection is challenging when analyzing tail data with a generalized Pareto distribution. Data below the threshold was not used in the model, resulting in incomplete characterization of the whole data. This paper applied the Gamma distribution, Weibull distribution, and lognormal distribution to fit the central data separately, and a generalized Pareto distribution (GPD) was used to analyze the tail data. In such composite models, the thresholds are estimated directly as parameters. We proposed an empirical distribution function-based parameter estimation method. The absolute value of the difference between the empirical distribution function and the composite distribution function was used as a loss function to obtain an estimate of the parameter. This parameter estimation method is suitable for complex multiparameter distributions. The estimation method based on the empirical distribution function was verified to be feasible through simulation studies. The composite model and the estimation method based on the empirical distribution function were applied to study the earthquake magnitude data to provide a reference for earthquake hazard analysis.

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