A Semi-Parametric KDE-GPD Model for Earthquake Magnitude Analysis

Abstract

A semi-parametric mixture model, combining kernel density estimation (KDE) and the generalized Pareto distribution (GPD), is applied to analyze the statistical characteristics of earthquake magnitudes. Data below a threshold are fitted using KDE, while data above the threshold are modeled using the GPD. Both the kernel bandwidth and the threshold are directly estimable as parameters. An estimation method based on the empirical distribution function (EDF) and maximum likelihood estimation (MLE) is used to estimate the parameters of the mixture model. The application of this model to earthquake magnitude analysis offers insights for seismic hazard assessment.

1. Introduction

The statistical patterns of earthquake magnitudes constitute a critical aspect in understanding seismic activity trends. Such research helps elucidate crustal movement dynamics while enhancing the accuracy and reliability of earthquake forecasting. The study of magnitude statistics has a long developmental history.

Gutenberg and Richter [1] first systematically analyzed observed earthquake magnitudes, noting that magnitude-frequency relationships generally follow specific patterns within regions exceeding a certain size. Given the destructive potential of major earthquakes, early research primarily focused on large seismic events, employing the GPD from extreme value theory to characterize the tail distributions of magnitude data [2,3,4]. However, the GPD only utilizes data exceeding a threshold while neglecting sub-threshold observations, whose distributional characteristics influence threshold selection and parameter estimation.

Subsequent improvements introduced parametric mixture models [5,6,7]. Zhang et al. [8] applied parametric mixture models to magnitude data analysis using EDF-based estimation, obtaining complete statistical distributions of seismic magnitudes. Parametric mixture models are widely adopted due to their interpretability, computational efficiency, and well-established theoretical properties. Nevertheless, like all parametric statistical inferences, they rely on strong model assumptions that may not hold in practice, potentially leading to erroneous conclusions.

Seminal studies pioneered the integration of parametric and nonparametric distributions to develop semi-parametric mixture models. Heckman and Singer [9] conducted comparative analyses of MLE across parametric mixture models and their semi-parametric counterparts. Wang and Chee [10] present a general framework for univariate non-parametric density estimation, based on mixture models with improved accuracy. Fienberg et al. [11] established a semi-parametric mixture logit regression framework capable of capturing “risk perception” through dichotomous responses, while Follmann and Lambert [12] subsequently employed semi-parametric mixture logistic regression to address overdispersion relative to binomial models. Davies [13] demonstrated the application in economic sectors, catalyzing expanded research interest in nonparametric and semi-parametric mixture approaches.

Substantial methodological advancements emerged in subsequent decades. Hall and Zhou [14] and Hall et al. [15] extended the framework to multivariate settings, while Bordes et al. [16] and Hunter et al. [17] developed novel parameter estimation techniques. Song et al. [18] implemented semi-parametric mixture models for sequential clustering, and Bordes et al. [19] investigated their regression formulations. These models have gained widespread adoption across economics, finance, biology, and medicine. Parallel progress occurred in estimation methodologies [20,21,22,23]. Young and Hunter [24] and Huang and Yao [25] examined proportionally varying semi-parametric mixture regression models, with the latter establishing convergence properties of the EM algorithm through smoothed likelihood theory. MacDonald et al. [26] constructed flexible mixture extreme-value models incorporating Bayesian estimation, while Pommeret and Vandekerkhove [27] systematically articulated the theoretical advantages of semi-parametric approaches. Contemporary applications include high-dimensional clustering [28] and penalized semi-parametric density estimation [29]. Martins-Ferreira et al. [30] introduced a hybrid framework integrating transformer neural networks with GPD to model compound flood exceedances. Chen and Zhang [31] proposed an adaptive Bayesian framework for threshold selection in extreme value mixture models. Vinayan et al. [32] employed Generalized Extreme Value (GEV) and GPD within an extreme value analysis framework to quantify variability in design wave heights. Liu and Zhou [33] developed an accelerated algorithm imposing constraints via penalized MLE.

Building upon this foundation, this study applies a semi-parametric composite model integrating KDE and GPD to analyze earthquake magnitudes. The model features direct estimation of kernel bandwidth and threshold as parameters, with estimation performed using MLE and an EDF-based approach. The application of magnitude analysis provides insights for seismic hazard assessment. The paper is structured as follows: Section 2 establishes the semi-parametric mixture model, Section 3 details parameter estimation, and Section 4 presents the simulation studies. Section 5 applies the model to evaluate seismic hazards in the eastern Bayan Har block, and Section 6 discusses conclusions and future directions.

2. Semi-Parametric Mixture Model

A general mixture model can be expressed as:

$$f(x)=\pi f_1(x)+(1-\pi )f_2(x)$$

(1)

Since extreme events are typically characterized by analyzing the tail behavior of data exceeding a high threshold, we model the upper tail using the GPD [34,35]. The sub-threshold regime, whose distributional form is not parametrically specified, is modeled through a non-parametric probability density function (PDF). Common methods for non-parametric PDF estimation include KDE, the maximum entropy method [36], Edgeworth series expansion [37], and orthogonal polynomial expansion. Comparative studies [38] demonstrate KDE’s superior versatility and reduced estimation errors [39,40]. Therefore, we adopt KDE as our fitting approach. In Equation (1), we set $\pi =1-{\varphi }_u$:

$$f_1(x)={\displaystyle \frac{h(x|\lambda ,X)}{H(u|\lambda ,X)}}{I}_{(-\infty ,u)}(x),f_2(x)=g(x|u,{\sigma }_u,\xi )I_{(u,\infty )}(x).$$

Let ${\varphi }_u=P(X>u)$, where ${\varphi }_u$ represents the proportion of data exceeding the threshold. This parameter quantifies the relative weights between KDE and GPD components in the mixture model. We can get

$$P(X>x)={\varphi }_u[1-P(X<x|X>u)]$$

If $X_1,X_2,\dots ,X_n$ is a sequence of independent and identically distributed (i.i.d.) random variables, the cumulative distribution function (CDF) of the semi-parametric mixture models [41] can be expressed as:

$$F(x|\lambda ,u,{\sigma }_u,\xi ,X)=\left\{\begin{array}{ll}(1-{\varphi }_u){\displaystyle \frac{H(x|\lambda ,X)}{H(u|\lambda ,X)}}& x\le u\\ (1-{\varphi }_u)+{\varphi }_{u}G(x|u,{\sigma }_u,\xi )& x>u\end{array}\right.$$

(2)

where $F(x|\lambda ,u,{\sigma }_u,\xi ,X)$ satisfies the continuity constraint

$$\underset{x\to u^{-}}{\lim }(1-{\varphi }_u){\displaystyle \frac{H(x|\lambda ,X)}{H(u|\lambda ,X)}}=\underset{x\to u^{+}}{\lim }(1-{\varphi }_u)+{\varphi }_{u}G(x|u,{\sigma }_u,\xi )=1-{\varphi }_u$$

The corresponding PDF is given by:

$$f(x|\lambda ,u,{\sigma }_u,\xi ,X)=\left\{\begin{array}{ll}(1-{\varphi }_u){\displaystyle \frac{h(x|\lambda ,X)}{H(u|\lambda ,X)}}& x\le u\\ {\varphi }_{u}g(x|u,{\sigma }_u,\xi )& x>u\end{array}\right.$$

(3)

where

$$G(x|u,{\sigma }_u,\xi )=P(X<x|X>u)=\left\{\begin{array}{ll}1-{[1+\xi ({\displaystyle \frac{x-u}{{\sigma }_u}})]}^{-{\displaystyle \frac{1}{\xi }}}& \xi \ne 0\\ 1-\exp [-({\displaystyle \frac{x-u}{{\sigma }_u}})]& \xi =0\end{array}\right.$$

(4)

$$g(x|u,{\sigma }_u,\xi )=\left\{\begin{array}{ll}{\displaystyle \frac{1}{{\sigma }_u}}{[1+\xi ({\displaystyle \frac{x-u}{{\sigma }_u}})]}^{-{\displaystyle \frac{1}{\xi }}-1}& \xi \ne 0\\ {\displaystyle \frac{1}{{\sigma }_u}}\exp [-({\displaystyle \frac{x-u}{{\sigma }_u}})]& \xi =0\end{array}\right.$$

(5)

represent the PDF and CDF of the GPD, respectively.

In the semi-parametric mixture PDF (3), the univariate KDE for $h(x|\lambda ,X)$ is defined as follows:

$$\widehat{h}(x|\lambda ,X)={\displaystyle \frac{1}{n\lambda }}{\displaystyle \sum _{i=1}^{n}K(\frac{x-x_i}{\lambda })}$$

where $K(x)$ is the kernel function, which typically satisfies the following conditions:

$$K(x)\ge 0,{\displaystyle {\int }_{-\infty }^{\infty }K(x)dx=1}\mathrm{and}\left\{\begin{array}{l}{\displaystyle \int xK(x)dx=0}\\ {\displaystyle \int x^{2}K(x)dx=c>0}\end{array}\right.$$

where $c$ is a constant. Many functions satisfy these conditions, including Gaussian and polynomial kernels. Research demonstrates, however, that fitting errors vary only slightly across different kernel selections, indicating little influence of kernel choice on non-parametric KDE accuracy. We therefore adopt the Gaussian kernel for probability density estimation in subsequent analysis, i.e., $K(x)={\displaystyle \frac{1}{\sqrt{2\pi }}}\exp (-{\displaystyle \frac{x^2}{2}})$. Then, $h(x|\lambda ,X)$ can be expressed as:

$$\widehat{h}(x|\lambda ,X)={\displaystyle \frac{1}{\sqrt{2\pi }n\lambda }}{\displaystyle \sum _{i=1}^n\exp \{-\frac{1}{2}{(\frac{x-x_i}{\lambda })}^2\}}$$

(6)

3. Parameter Estimation of the Semi-Parametric Mixture Model

3.1. MLE of Parameters

Due to the complexity of the likelihood function, analytical solutions are infeasible. Therefore, we compute MLE numerically. The semi-parametric mixture model’s likelihood function is:

$$L(\theta |X)=L_{K\mathrm{DE}}(\lambda ,u|X)L_{GPD}(u,\mu ,\sigma ,\xi |X)$$

Habbema et al., 1974 and Duin (1976) [42,43] pioneered likelihood-based bandwidth estimation for KDE. The KDE likelihood function is given by:

$$L_{KDE}(\lambda ,X)={\displaystyle \prod _{x_i\le u}\frac{(1-{\varphi }_u)}{H(u|\lambda ,X)}\frac{1}{n\lambda }}{\displaystyle \sum _{j=1}^{n}K(\frac{x_i-x_j}{\lambda })}$$

However, when $x_i=x_j$, (so $x_i-x_j=0$), and $\lambda \to 0$, this result in a degenerate likelihood function [43]. To address this issue, the likelihood function can be replaced with a cross-validation likelihood function:

$$L_{K\mathrm{DE}}(\lambda ,u|X)={\displaystyle \prod _{x_i\le u}\frac{(1-{\varphi }_u)}{H(u|\lambda ,X)}\frac{1}{(n-1)\lambda }}{\displaystyle \sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^{n}K(\frac{x_i-x_j}{\lambda })}$$

(7)

Equation (7) can be interpreted as minimizing the Kullback–Leibler divergence (K-L distance) [44,45]. Let $A=\{x_i\le u\}$ and $B=\{x_i>u\}$, where the samples belong in the PDF. The kernel density likelihood function can be expressed as a function of the bandwidth.

$$L_{KDE}(\lambda ,u|X)={\left\{{\displaystyle \frac{(1-{\varphi }_u)}{\frac{1}{n}{\displaystyle \sum _{i=1}^n\mathsf{\Phi }(\frac{u-x_i}{\lambda })}}}\right\}}^{|A|}{\displaystyle \prod _A\frac{1}{(n-1)\lambda }}{\displaystyle \sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^{n}K(\frac{x_i-x_j}{\lambda })}$$

(8)

The tail portion of the sample follows a GPD, and the corresponding likelihood function is given by:

$$L_{GPD}(u,\sigma ,\xi |X)=\left\{\begin{array}{l}{\displaystyle \prod _{\{i:x_i>u\}}\frac{1}{\sigma }{(1+\xi \frac{x_i-u}{\sigma })}^{-\frac{1}{\xi }-1},\xi \ne 0}\\ {\displaystyle \prod _{\{i:x_i>u\}}\frac{1}{\sigma }\exp (\frac{x_i-\mu }{\sigma }),\xi =0}\end{array}\right.$$

(9)

The likelihood function for the semi-parametric model is given by:

$$\begin{gathered} L(\theta |X)=L_{K\mathrm{DE}}(\lambda ,u|X)L_{GPD}(u,\mu ,\sigma ,\xi |X) \\ =\left\{\begin{array}{l}{\left({\displaystyle \frac{(1-{\varphi }_u)}{\frac{1}{n}{\displaystyle \sum _{i=1}^n\varphi (\frac{u-x_i}{\lambda })}}}\right)}^{|A|}{\displaystyle \prod _A\frac{1}{(n-1)}}{\displaystyle \sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^n{K}_{\lambda }(x_i-x_j)}{\displaystyle \prod _B\frac{1}{\sigma }{(1+\xi \frac{x_i-\mu }{\sigma })}^{-\frac{1}{\xi }-1},\xi \ne 0}\\ {\left({\displaystyle \frac{(1-{\varphi }_u)}{\frac{1}{n}{\displaystyle \sum _{i=1}^n\varphi (\frac{u-x_i}{\lambda })}}}\right)}^{|A|}{\displaystyle \prod _A\frac{1}{(n-1)}}{\displaystyle \sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^n{K}_{\lambda }(x_i-x_j)}{\displaystyle \prod _B\frac{1}{\sigma }\exp (\frac{x_i-\mu }{\sigma }),\xi =0}\end{array}\right. \end{gathered}$$

(10)

The corresponding log-likelihood function is expressed as:

$$\begin{gathered} \ln L(\theta |X)=\ln L_{K\mathrm{DE}}(\lambda ,u|X)+\ln L_{GPD}(u,\mu ,\sigma ,\xi |X) \\ =\left\{\begin{array}{l}|A|\ln \left({\displaystyle \frac{1-\varphi (u)}{\frac{1}{n}{\displaystyle \sum _{i=1}^n\mathsf{\Phi }(\frac{u-x_i}{\lambda })}}}\right)+|A|\ln \left[{\displaystyle \frac{1}{(n-1)\lambda }}\right]+{\displaystyle \sum _A[\ln {\displaystyle \sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^n\phi (\frac{x-x_j}{\lambda })]}+}{\displaystyle \sum _B[-\ln \sigma +(\frac{-1}{\xi }-1)\ln (1+\xi \frac{x_i-\mu }{\sigma })],\xi \ne 0}\\ A|\ln \left({\displaystyle \frac{1-\varphi (u)}{\frac{1}{n}{\displaystyle \sum _{i=1}^n\mathsf{\Phi }(\frac{u-x_i}{\lambda })}}}\right)+|A|\ln \left[{\displaystyle \frac{1}{(n-1)\lambda }}\right]+{\displaystyle \sum _A[\ln {\displaystyle \sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^n\phi (\frac{x-x_j}{\lambda })]}+}{\displaystyle \sum _B(-\ln \sigma +\frac{x_i-\mu }{\sigma }),\xi \ne 0}\end{array}\right. \end{gathered}$$

(11)

3.2. Parameter Estimation Method Based on the EDF

Assume $F_n(x)$ denotes the sample EDF. We define a loss function

$$Loss=|F(x)-F_n(x)|,$$

whose minimizer yields EDF-based parameter estimates. Since analytical minimization is infeasible, we use the Newton–Raphson method for numerical optimization.

4. Simulation of Parameter Estimation for Semi-Parametric Mixture Models

To evaluate the semi-parametric model’s performance, we generated random samples from three parametric mixture distributions:

• Normal (shape = 1, scale = 4) + GPD.
• Weibull (shape = 1.5, scale = 2) + GPD.
• Gamma (shape = 1, scale = 2) + GPD.

All cases shared common GPD parameters (shape parameter: −0.2, scale parameter: 1.5). The semi-parametric mixture model was fitted to each sample, with parameter estimates compared against true values.

Table 1. Parameter estimates for semi-parametric model (Normal + GPD).

$\mathit{N}(1,2^2)+\mathbf{GPD}(\mathit{\xi }=-0.2,\mathit{\sigma }=1.5)$	$\widehat{\mathit{\lambda }}$	$\widehat{\mathit{u}}$	$\widehat{\mathit{\xi }}$	$\widehat{\mathit{\sigma }}$
Parameter estimations by EDF	0.6005	2.0153	−0.2358	1.4802
Standard deviation	0.0052	0.0211	0.0498	0.0001
Confidence interval (confidence level $\alpha =0.05$)	(0.6000, 0.6001)	(1.752, 2.0664)	(−0.3750, −0.0589)	(1.3800, 1.5816)
Parameters by MLE	0.3817	2.0053	−0.2388	1.4901
Standard deviation	0.1213	0.0142	0.0388	0.0008
Confidence interval (confidence level $\alpha =0.05$)	(0.1035, 0.5811)	(1.300, 2.1438)	(−0.3419, 0.1207)	(1.331, 1.6734)

presents the parameter estimates, standard deviations, and 95% confidence intervals for the normal-GPD case.

Figure 1 compares the EDF-estimated CDF with the EDF of the simulated data. Figure 2 displays the simulated data histogram alongside the semi-parametric mixture model’s fitted PDF using EDF-estimated parameters.

Figure 3 compares the MLE-estimated CDF with the EDF of the simulated data.

Figure 4 displays the simulated data histogram alongside the semi-parametric mixture model’s fitted PDF using MLE-estimated parameters. As shown in Table 1 and Figure 1, Figure 2, Figure 3 and Figure 4, both parameter estimation methods yield acceptable results for the normal-GPD mixture data.

Table 2. Parameter estimates for sesemi-parametric model (Weibull + GPD).

$\mathbf{Weibull}(1.5,2)+\mathbf{GPD}(\mathit{\xi }=-0.2,\mathit{\sigma }=1.5)$	$\widehat{\mathit{\lambda }}$	$\widehat{\mathit{u}}$	$\widehat{\mathit{\xi }}$	$\widehat{\mathit{\sigma }}$
Parameter estimations by EDF	0.6235	2.0003	−0.1958	1.9802
Standard deviation	0.1241	0.0002	0.0498	0.1950
Confidence interval (confidence level $\alpha =0.05$)	(0.1003, 0.8005)	(1.8725, 2.0346)	(0.2975, 0.1018)	(1.2943, 2.2473)
Parameters by MLE	0.7482	2.003	−0.2080	1.7744
Standard deviation	0.1774	0.0142	0.0247	0.0877
Confidence interval (confidence level $\alpha =0.05$)	(0.1203, 0.8011)	(2.0064, 2.2438)	(−0.3419, −0.1007)	(1.3351, 1.8281)

presents the parameter estimates, standard deviations, and 95% confidence intervals for the Weibull-GPD case.

Figure 5 compares the EDF-estimated CDF with the EDF of the simulated data. Figure 6 displays the simulated data histogram alongside the semi-parametric mixture model’s fitted PDF using EDF-estimated parameters.

Figure 7 compares the MLE-estimated CDF with the EDF of the simulated data. Figure 8 shows the simulated data histogram alongside the semi-parametric mixture model’s fitted PDF with MLE-estimated parameters.

Table 2 and Figure 5, Figure 6, Figure 7 and Figure 8 demonstrate that both parameter estimation methods yield acceptable results for the Weibull-GPD composite data.

Table 3. Parameter estimates for semi-parametric model (Gamma + GPD).

$\mathbf{Gamma}(1.5,2)+\mathbf{GPD}(\mathit{\xi }=-0.2,\mathit{\sigma }=1.5)$	$\widehat{\mathit{\lambda }}$	$\widehat{\mathit{u}}$	$\widehat{\mathit{\xi }}$	$\widehat{\mathit{\sigma }}$
Parameter estimations by EDF	0.1001	2.0051	−0.2463	1.4901
Standard deviation	0.0005	0.0124	0.0235	0.0430
Confidence interval (confidence level $\alpha =0.05$)	(0.1001, 0.80121)	(1.5764, 2.5995)	(−0.2803, 0.1484)	(1.3802, 1.5682)
Parameters by MLE	0.1072	2.0235	−0.2176	1.4880
Standard deviation	0.0141	0.0507	0.0683	0.0295
Confidence interval (confidence level $\alpha =0.05$)	(0.1103, 0.6012)	(1.7003, 2.3644)	(−0.1504, −0.0146)	(1.3213, 1.6180)

presents the parameter estimates, standard deviations, and 95% confidence intervals for the Gamma-GPD case.

Figure 9 compares the EDF-estimated CDF with the EDF of the simulated data. Figure 10 displays the simulated data histogram alongside the semi-parametric mixture model’s fitted PDF with EDF-estimated parameters.

Figure 11 compares the MLE-estimated CDF estimated with the EDF of the simulated data. Figure 12 presents the simulated data histogram alongside the semi-parametric mixture model’s fitted PDF with the MLE-estimated parameters.

Table 3 and Figure 9, Figure 10, Figure 11 and Figure 12 show that both parameter estimation methods yield acceptable results for the Gamma-GPD mixture data.

Simulation results across all scenarios confirm the semi-parametric mixture model’s strong performance. We next apply it to real seismic data analysis.

5. Statistical Characteristic Analysis of Seismic Magnitude Data in the Eastern Bayan Har Block

Earthquake magnitude data were obtained from the National Seismic Science Data Sharing Center (https://data.earthquake.cn/ (accessed on 29 April 2019)). After removing aftershocks using the C-S method [46], 19,221 records before December 2019 were retained as research samples. Figure 13 displays the annual magnitude histogram.

5.1. Data Statistics

Table 4. Statistical summary of seismic magnitude data.

Minimum	Maximum	Mean	First Quartile (Q1)	Third Quartile (Q3)	Variance	Skewness	Kurtosis
0.0500	8.1000	0.9617	0.3900	1.4100	0.6575	1.5523	6.7771

summarizes key statistics: a kurtosis of 6.7771 indicates leptokurtic distribution, while a positive skewness (1.5523) confirms right-skewed data with a heavy upper tail. This occurs when infrequent large values extend the distribution’s right tail.

5.2. Nonparametric KDE of the Data

Using a normal kernel function for KDE, we computed the optimal bandwidth via Equation (6). Figure 14 superimposes the KDE curve on the data histogram, demonstrating close CDF alignment. Note that while KDE provides an excellent visual correspondence, its lack of explicit PDF expression may constrain advanced applications.

5.3. Data Fitting Using the Semi-Parametric Mixture Model

This section applies the semi-parametric mixture model established in Section 3 to seismic magnitude data. We obtained the MLE of parameters by maximizing the log-likelihood function using MATLAB’s (R2018b) optimization algorithms. This method simultaneously estimates the bandwidth as a parameter, eliminating errors from manual selection.

Table 5. MLEs of parameters for nonparametric and semi-parametric models.

Models	$\widehat{\mathit{\lambda }}$	$\widehat{\mathit{u}}$	$\widehat{\mathit{\xi }}$	$\widehat{\mathit{\sigma }}$
KDE	0.0591
Semi-parametric mixture model (KDE-GPD)	1.0001	4.9801	−0.2021	0.7514

presents parameter estimates for both the non-parametric KDE and the semi-parametric (KDE-GPD) mixture model.

Figure 15 shows the semi-parametric mixture model’s PDF. The blue segment depicts the KDE component for sub-threshold data, while the red segment represents the GPD tail model for supra-threshold values.

Figure 16 compares the fitted EDF with the actual data EDF, showing consistent trends.

We performed a Kolmogorov–Smirnov (K-S) goodness-of-fit test for the tail data’s conformance to GPD, obtaining a p-value of 0.4665. This fails to reject the null hypothesis, supporting the GPD assumption’s statistical plausibility. Additionally, Figure 17 further validates model fit through a Q-Q plot showing alignment between empirical quantiles and theoretical GPD quantiles along the diagonal.

Using the formula ${\widehat{x}}^{*}=\widehat{u}-{\displaystyle \frac{\widehat{\sigma }}{\widehat{\xi }}}$, we calculate a maximum magnitude of Ms 8.73. This estimate aligns with historical earthquakes in the region: the 2001 Kunlun Mountain Pass (Ms 8.0), 2008 Wenchuan (Ms 8.0), 2010 Yushu (Ms 7.1), 2013 Lushan (Ms 7.0), 2021 Qinghai Maduo (Ms 7.4), and 2022 Luding (Ms 6.8).

For tail data characterization, quantile estimation provides critical seismic hazard indicators. We compute return levels for specified return periods using $T=1/(1-p)$, which is also a key indicator of seismic hazard. The quantile is calculated by the formula $x_p=u+{\displaystyle \frac{\sigma }{\xi }}(p^{-\xi }-1)$. The estimated return level for return period $T=1/(1-p)$ is:

$${\widehat{x}}_{1-{\displaystyle \frac{1}{365T}}}=\widehat{u}+{\displaystyle \frac{\widehat{\sigma }}{\widehat{\xi }}}\left({\left({\displaystyle \frac{n}{N_u}}(1-p)\right)}^{-\widehat{\xi }}-1\right)=\widehat{u}+{\displaystyle \frac{\widehat{\sigma }}{\widehat{\xi }}}\left({\left({\displaystyle \frac{365T\cdot N_u}{n}}\right)}^{\widehat{\xi }}-1\right)$$

(12)

Table 6. Return levels for different return periods.

Return Period (Years)	Return Level
10	6.3635
20	6.6687
30	6.8283
50	7.0117
60	7.0727
80	7.1645
100	7.2322

presents the return levels for different return periods, computed from Equation (12).

The return level plot (Figure 18) demonstrates strong model–data alignment, with empirical return levels within 95% confidence intervals. However, extrapolation to very high return levels involves substantial uncertainty.

6. Conclusions

Semi-parametric mixture models balance flexibility, efficiency, and interpretability by integrating parametric and non-parametric approaches. Our application to magnitude data pioneers new uses of these models: the sub-threshold portion uses non-parametric fitting, while supra-threshold data follows GPD, enabling complete statistical characterization of seismic magnitudes. Both threshold and bandwidth are directly estimated as parameters, eliminating subjectivity in conventional threshold selection methods. While LOOL-enhanced KDE improves accuracy, its computational demands highlight the need for efficiency-optimized algorithms that preserve precision, a key advancement target. Future research should address smooth transitions between nonparametric and parametric components, incorporate spatial dependencies, and account for temporal non-stationarity in complete magnitude catalogs.