Generalized Pareto Distribution Research Articles

Employing the generalized Pareto distribution to analyze extreme rainfall events on consecutive rainy days in Thailand's Chi watershed: implications for flood management

Abstract. Extreme rainfall events in the Chi watershed of northeastern Thailand have significant implications for the safe and economic design of engineered structures and effective reservoir management. This study investigates the characteristics of extreme rainfall events in the watershed and their implications for flood risk management. We apply extreme value theory to historical maximum cumulative rainfall data for consecutive rainy days from 1984 to 2022. The generalized Pareto distribution (GPD) was used to model the extreme rainfall data, with the parameters estimated using maximum likelihood estimation (MLE) and linear moment estimation (L-ME) methods based on specific conditions. The goodness-of-fit tests confirm the suitability of the GPD for the data, with p values exceeding 0.05. Our findings reveal that certain regions, notably Udon Thani, Chaiyaphum, Maha Sarakham, Tha Phra Agromet., Roi Et, and Sisaket provinces, show the highest return levels for consecutive 2 d (CONS-2) and 3 d (CONS-3) rainfall. These results underscore the heightened risk of flash flooding in these regions, even with short periods of continuous rainfall. Based on our findings, we developed 2D return level maps using the Q-geographic information system (Q-GIS) program, providing a visual tool to assist with flood risk management. The study offers valuable insights for designing effective flood management strategies and highlights the need for considering extreme rainfall events in water management and planning. Future research could extend our findings through spatial correlation analysis and the use of copula functions. Overall, this study emphasizes the importance of preparing for extreme rainfall events, particularly in the era of climate change, to mitigate potential flood-related damage.

Estimation of Intensity‐Duration‐Area‐Frequency Relationships Based on the Full Range of Non‐Zero Precipitation From Radar‐Reanalysis Data

AbstractIntensity‐Duration‐Area‐Frequency (IDAF) models provide the mathematical link between precipitation intensities (I), durations (D), areas (A), and frequency of occurrence (F). They play a critical role in hydrological design, areal rainfall hazard quantification, storm characterization, and early warning system development. IDAF models extend the conventional Intensity‐Duration‐Frequency models by accounting for the spatial extent of precipitation (i.e., the area). In this study, we develop IDAF models using the entire non‐zero precipitation intensities, not only the extremes. We use the extended generalized Pareto distribution (EGPD) to model the precipitation intensities. To build the IDAF models, we adopt a data‐driven approach that allows the linkage of EGPD parameters with duration and area, based on empirically determined parametric relationships. The inference of model parameters is done using a global maximum likelihood estimation, and uncertainties are assessed by the bootstrap method. The study area is Switzerland, a topographically complex region of 42,000 km2 with regional precipitation variability and clear seasonality. The study utilizes 17 years of data from CombiPrecip, a radar‐reanalysis product developed by geostatistically merging radar and rain gauge data in an operational setting. We build the IDAF models for the spatiotemporal range of 1–72 hr and 1 to 1,089 km2 at each pixel in the study area. To the best of our knowledge, our study is the first attempt to use the EGPD in IDAF curve modeling. It discusses the use and limitations of CombiPrecip in extreme value analysis and highlights the challenges of modeling areal precipitation in a complex topographical environment.

Modeling Extreme Precipitation Data in a Mining Area

In recent decades, extreme precipitation events have increased in frequency and intensity in Greece and across regions of the Mediterranean, with significant environmental and socioeconomic impacts. Therefore, extensive statistical analysis of the extreme rainfall characteristics on a dense temporal scale is crucial for areas with important economic activity. For this reason, this paper uses the daily precipitation measurements of four meteorological stations in a mining area of northeastern Chalkidiki peninsula from 2006 to 2021. Three statistical approaches were carried out to develop the best-fitting probability distribution for annual extreme precipitation conditions, using the maximum likelihood method for parameter estimation: the block maxima of the generalized extreme value (GEV) distribution and the peak over threshold of the generalized Pareto distribution (GPD) based on extreme value theory (EVT), and the gamma distribution. Based upon this fitting distribution procedure, return periods for the extreme precipitation values were calculated. Results indicate that EVT distributions satisfactorily fit extreme precipitation, with GPD being the most appropriate, and lead to similar conclusions regarding extreme events.

Bayesian Sensitivity Analysis for VaR and CVaR Employing Distorted Band Priors

In the context of robust Bayesian analysis, studies mainly focus on computing the range of some quantities of interest when the prior distribution varies in a class. We use the concept of distorted bands to introduce a family of priors on the shape parameter of the Generalized Pareto distribution. We show how certain properties of the likelihood ratio order allow us to propose novel sensitivity measures for Value at Risk and Conditional Value at Risk, which are the most useful and reliable risk measures. Although we focus on the Generalized Pareto distribution, which is essential in Extreme Value Theory, the new sensitivity measures could be employed for all the distributions that verify certain conditions related to likelihood ratio order. A thorough simulation study was carried out to perform a sensitivity analysis, and two illustrative examples are also provided.

A Novel Load Extrapolation Method for Multiple Non-Stationary Loads on the Drill Pipe of a Rotary Rig

The drill pipe of a rotary rig is subject to the dynamic influence of non-stationary loads, including rotation torque and applied force. In order to address the challenge of simultaneously extrapolating multiple non-stationary loads, a novel extrapolation framework is proposed. This framework utilizes rainflow counting to obtain mean and amplitude sequences of the loads. The extreme values of the amplitude sequence are fitted using the Generalized Pareto Distribution (GPD), while the median values are fitted using the Double Kernel Density Estimation (DKDE). By extrapolating the Inverse Cumulative Distribution Function (ICDF) based on the fitted distribution, a new amplitude sequence can be derived. The combination of this extrapolated amplitude sequence with the original mean sequence forms a new load spectrum. The results of applying the proposed extrapolation method to the drill pipe of a rotary rig demonstrate the ability of the method to yield conservative extrapolation results and accurately capture the variations in damage under the original working conditions.

Gini-stable Lorenz curves and their relation to the generalised Pareto distribution

We introduce an iterative discrete information production process where we can extend ordered normalised vectors by new elements based on a simple affine transformation, while preserving the predefined level of inequality, G, as measured by the Gini index.Then, we derive the family of empirical Lorenz curves of the corresponding vectors and prove that it is stochastically ordered with respect to both the sample size and G which plays the role of the uncertainty parameter. We prove that asymptotically, we obtain all, and only, Lorenz curves generated by a new, intuitive parametrisation of the finite-mean Pickands' Generalised Pareto Distribution (GPD) that unifies three other families, namely: the Pareto Type II, exponential, and scaled beta distributions. The family is not only totally ordered with respect to the parameter G, but also, thanks to our derivations, has a nice underlying interpretation. Our result may thus shed a new light on the genesis of this family of distributions.Our model fits bibliometric, informetric, socioeconomic, and environmental data reasonably well. It is quite user-friendly for it only depends on the sample size and its Gini index.

Statistical characteristics of earthquake magnitude based on the composite model

<abstract><p>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.</p></abstract>

Generalized Pareto distribution and income inequality: an extension of Gibrat's law

<abstract> <p>Motivated by empirical observations, we proposed a possible extension of Gibrat's law. By applying it into the random growth theory of income distribution, we found that the income distribution is described by a generalized Pareto distribution (GPD) with three parameters. We observed that there is a parameter $ \eta $ in the GPD that plays a key role in determining the shape of income distribution. By using the Kolmogorov-Smirnov test, we empirically showed that, for typical market-economy countries, $ \eta $ is significantly close to 0, indicating that the income distribution is characterized by a two-class pattern: The bottom 90% of the population is approximated by an exponential distribution, while the richest 1%~3% is approximated by an asymptotic power law. However, we empirically found that in China during the period of the planned economy and the early stages of market reform (from 1978 to 1990), $ \eta $ deviated significantly from 0, indicating that the bottom of the population no longer conformed to an exponential distribution.</p> </abstract>

Discrete Extremes

Our contribution is to widen the scope of extreme value analysis applied to discrete-valued data. Extreme values of a random variable are commonly modeled using the generalized Pareto distribution, a peak-over-threshold method that often gives good results in practice. When data is discrete, we propose two other methods using a discrete generalized Pareto and a generalized Zipf distribution respectively. Both are theoretically motivated and we show that they perform well in estimating rare events in several simulated and real data cases such as word frequency, tornado outbreaks and multiple births.

On Alpha Power Transformation Generalized Pareto Distribution and Some Properties

Recently, the need to develop statistical distributions has become the most important spot. In this context, we employ the α‐power transformation (APT) method to convert the generalized Pareto distribution (GPD) into a new distribution. Some statistical properties of the proposed distribution are being studied, such as moments, arithmetic mean, moment‐generating function, random variables, entropy, reliability, and hazard function (HF). In addition, the proposed distribution is compared with the Pareto distribution and some other forms of alpha power distributions, such as the alpha power Pareto (APP) distribution, alpha power Rayleigh (APR) distribution, and the alpha power Lomax (APL) distribution. Finally, we demonstrate the benefits of the proposed distribution through a simulation study and two real data sets. It was found that the results showed the MLE method is reliable, the APTGP distribution is a competitive distribution for the aforementioned data set, and it is a mirror image of the Pareto distribution.

THE VAR EVALUATION OF SHARIAH STOCK MARKET IN MALAYSIA DURING COVID-19 PANDEMIC BY USING CONDITIONAL EVT METHOD

In the current financial market, the Islamic stock market faced with a significant challenge to sustain and maintain its stability in intensified market volatility and unexpected extreme events. It can reduce the intensity and occurrence of financial crises by eliminating the primary vulnerabilities of the conventional system. This paper aims to identify the most effective method in risk evaluation by presenting the risk evaluation performance between conventional and Islamic stock market that focusing on extreme events in stock market returns. The data analysis is divided into two periods: normal and crisis COVID-19 periods. The empirical analysis, conducted within the sample employs the conditional extreme value theory (EVT) method that combine the filtered series of GARCH and EGARCH models. This filtered series will be used to generate the threshold by using the peak-over-threshold (POT) method. This threshold then will be used to estimate the generalized Pareto distribution (GPD) distribution to forecast the one-day ahead value-at-risk (VaR). The findings indicate that, in Shariah stock markets, the conditional EVT model demonstrates superior performance in forecasting stock market risk compared to the standard GARCH and EGARCH models.

Distributions and extreme value analysis of critical response rate and split-time metric in nonlinear oscillators with stochastic excitation

A single-degree-of-freedom oscillator with a nonlinear restoring force and stochastic external forcing is studied with the goal of understanding the distributions of certain response rates. Motivated by applications to ship motions, the response rates concern the minimum response rate leading to capsizing (the critical response rate) and a measurement comparing an observed response rate to the critical response rate (the split-time metric), whose distributions are investigated both analytically and numerically. Three nonlinear restoring forces are considered: piecewise linear (experiencing linearly softening stiffness above a “knuckle” point), doubly piecewise linear (experiencing piecewise linearly softening stiffness above a “knuckle” point), and the cubic softening restoring force of the Duffing oscillator. In the first two cases, an efficient numerical simulation of the critical response rate and split-time metric is proposed from a derived distribution; in the latter case, the density of the critical response rate is approximated assuming white noise excitation. A key interest is in the nature of the right tail of the split-time metric, specifically as it relates to its extrapolation through extreme value analysis in estimating the probability of capsizing. The distribution is found to have a “light” tail, which motivates the use of exponential rather than the generalized Pareto distribution for exceedances above threshold in extreme value analysis. Finally, threshold selection through a prediction error criterion for the exponential distribution is examined, and the Weibull distribution tail is suggested as a useful means for a more refined examination of the distribution tail.

Return levels of extreme European windstorms, their dependency on the North Atlantic Oscillation, and potential future risks

Abstract. Windstorms are the most damaging natural hazard across western Europe. Risk modellers are limited by the observational data record to only ∼ 60 years of comprehensive reanalysis data that are dominated by considerable inter-annual variability. This makes estimating return periods of rare events difficult and sensitive to the choice of the historical period used. This study proposes a novel statistical method for estimating wind gusts across Europe based on observed windstorm footprints. A good description of extreme wind speeds is obtained by assuming that gust speed peaks over threshold are distributed exponentially, i.e. a generalised Pareto distribution having a zero shape parameter. The threshold and tail scale parameter are estimated at each location and used to calculate estimates of the 10- and 200-year return levels. The North Atlantic Oscillation (NAO) is particularly important for modulating lower return levels and modulating the threshold, with a less detectable influence on rarer extremes and the tail scale parameter. The length of historical data required to have the lowest error in estimating return levels is quantified using both observed and simulated time series of the historical NAO. For reducing errors in estimating 200-year return levels of an independent 10-year period, a data catalogue of at least 20 years is required. For lower return levels the NAO has a stronger influence on estimated return levels, and so there is more variability in estimates. Using theoretical estimates of future NAO states, return levels are largely outside the historical uncertainty, indicating significant increases in risk potential from windstorms in the next 100 years. Our method presents a framework for assessing high-return-period events across a range of hazards without the additional complexities of a full catastrophe model.

Extreme value analysis of wind droughts in Great Britain

Due to the growing proportion of wind energy in Great Britain’s energy mix, prolonged periods of low wind power generation have become a significant challenge for decarbonising the electricity system. As such, characterising drought severity and duration is important for ensuring the reliability of the electricity system. Employing concepts derived from hydrology, an extreme value analysis was carried out on wind drought events in Great Britain based on 72 years of ERA5 reanalysis data. The application of pooling procedures was found to be beneficial in robustly identifying wind droughts in cases where the capacity factor is not constantly below an arbitrary threshold. The sequent peak algorithm pooling was found to have particular relevance for electricity systems where energy storage technologies are used to compensate for low wind power generation. The Pearson-III distribution was identified as a suitable model to represent extreme wind droughts, while the Lognormal and Generalised Pareto distributions are also viable alternatives. Sustained periods of low wind power generation with a duration of 14 days were estimated to have a return period of five years and the longest event on record of approximately 26 days is expected to occur once every 100 years. The investigation of these wind droughts from a hydrological perspective has thus shown that they may not be particularly rare occurrences.

A Bayesian extreme value theory modelling framework to assess corridor-wide pedestrian safety using autonomous vehicle sensor data

Pedestrians are a vulnerable road user group, and their crashes are generally spread across the network rather than in a concentrated location. As such, understanding and modelling pedestrian crash risk at a corridor level becomes paramount. Studies on pedestrian crash risks, particularly with the traffic conflict data, are limited to single or multiple but scattered intersections. A lack of proper modelling techniques and the difficulties in capturing pedestrian interaction at the network or corridor level are two main challenges in this regard. With autonomous vehicles trialled on public roads generating massive (and unprecedented) datasets, utilising such rich information for corridor-wide safety analysis is somewhat limited where it appears to be most relevant. This study proposes an extreme value theory modelling framework to estimate corridor-wide pedestrian crash risk using autonomous vehicle sensor/probe data. Two types of models were developed in the Bayesian framework, including the block maxima sampling-based model corresponding to Generalised Extreme Value distribution and the peak over threshold sampling-based model corresponding to Generalised Pareto distribution. The proposed framework was applied to autonomous vehicle data from Argoverse—a Ford Motors subsidiary. This autonomous vehicle fleet of Agro AI (owner of Argoverse dataset) is equipped with two 64 beams synchronised LiDAR sensors, a cluster of seven high-resolution cameras, and a pair of stereo-vison high-resolution cameras to capture surrounding road users’ information within a range of 200 meters. A subset of the Argoverse dataset, focussing on an arterial corridor in Miami, USA, was used to extract pedestrian and vehicle trajectories. From these trajectories, vehicle–pedestrian conflicts were identified and measured using post encroachment time. The non-stationarity of extremes was captured by vehicle volume, pedestrian volume, average vehicle speed, and average pedestrian speed in the extreme value model. Both block maxima and peak over threshold sampling-based models were found to provide a reasonable estimate of historical pedestrian crash frequencies. Notably, the block maxima sampling-based model was more accurate than the peak over threshold sampling-based model based on mean crash estimates and confidence intervals. This study demonstrates the potential of using autonomous vehicle sensor data for network-level safety, enabling an efficient identification of pedestrian crash risk zones in a transport network.

Modeling Extreme COVID-19 Data in Iraq

This paper considers the maximum number of weekly cases and deaths caused by the COVID-19 pandemic in Iraq from its outbreak in February 2020 until the first of July 2022. Some probability distributions were fitted to the data. Maximum likelihood estimates were obtained and the goodness of fit tests were performed. Results revealed that the maximum weekly cases were best fitted by the Dagum distribution, which was accepted by three goodness of fit tests. The generalized Pareto distribution best fitted the maximum weekly deaths, which was also accepted by the goodness of fit tests. The statistical analysis was carried out using the Easy-Fit software and Microsoft Excel 2019.

Estimation of probability of large roll angle with envelope peaks over threshold method

The development and application of the Envelope Peak-over-Threshold (EPOT) method is summarized for the roll motion of a ship in irregular waves. The method is based on the conventional Peak-over-Threshold (POT) method, but uses an envelope of the peaks for declustering the data. As suggested by a reduced-order model, a power-law tail should be applicable for the peaks of roll motions, which allows the introduction of physical considerations into the statistical model. A data-driven Generalized Pareto distribution (GPD) is applied to study the behavior of the shape parameter on large-volume samples in order to confirm the theoretical prediction of the tail structure. The physics-informed statistical model approximates data above a suitable threshold with a Pareto distribution, while the threshold is determined with a prediction error criterion. An example is included, where extrapolations with GPD are compared to the physics-informed model.

Heavy-tailed distributions of confirmed COVID-19 cases and deaths in spatiotemporal space.

This paper conducts a systematic statistical analysis of the characteristics of the geographical empirical distributions for the numbers of both cumulative and daily confirmed COVID-19 cases and deaths at county, city, and state levels over a time span from January 2020 to June 2022. The mathematical heavy-tailed distributions can be used for fitting the empirical distributions observed in different temporal stages and geographical scales. The estimations of the shape parameter of the tail distributions using the Generalized Pareto Distribution also support the observations of the heavy-tailed distributions. According to the characteristics of the heavy-tailed distributions, the evolution course of the geographical empirical distributions can be divided into three distinct phases, namely the power-law phase, the lognormal phase I, and the lognormal phase II. These three phases could serve as an indicator of the severity degree of the COVID-19 pandemic within an area. The empirical results suggest important intrinsic dynamics of a human infectious virus spread in the human interconnected physical complex network. The findings extend previous empirical studies and could provide more strict constraints for current mathematical and physical modeling studies, such as the SIR model and its variants based on the theory of complex networks.

Shrinkage Methods for Estimating the Shape Parameter of the Generalized Pareto Distribution

The generalized Pareto distribution is one of the most important distributions in statistics of extremes as it has wide applications in fields such as finance, insurance, and hydrology. This study proposes two new methods for estimating the shape parameter of the generalized Pareto distribution (GPD). The proposed methods use the shrinkage principle to adapt the existing empirical Bayesian with data-based prior and the likelihood moment method to obtain two estimators. The performance of the proposed estimators is compared with the existing estimators (i.e., maximum likelihood, likelihood moment estimators, etc.) for the shape parameter of the generalized Pareto distribution in a simulation study. The results show that the proposed estimators perform better for small to moderate number of exceedances in estimating shape parameter of the light-tailed distributions and competitive when estimating heavy-tailed distributions. The proposed estimators are illustrated with practical datasets from climate and insurance studies.

Dynamic Bayesian hierarchical peak over threshold modeling for real-time crash-risk estimation from conflict extremes

Using traffic conflict-based extreme value theory (EVT) models to quantify real-time crash-risk of road facilities is a promising direction for developing proactive traffic safety management strategies. Existing EVT real-time crash-risk analysis studies have only focused on using block maxima models. This study proposes a dynamic Bayesian hierarchical peak over threshold modeling approach to estimate real-time crash-risk based on traffic conflicts. The proposed approach combines quantile regression, dynamic updating approach, Bayesian hierarchical structure, and the peak over threshold method to generate time-varying generalized Pareto distributions to derive real-time crash-risk measures (i.e., crash probability and return level). The derived real-time crash-risk measures are applied to estimate cycle-level crash-risk at three signalized intersections in Surrey, British Columbia. Five approaches are used to dynamically update the model parameters, including time trend model, generalized autoregressive conditional heteroskedasticity process approach, as well as the first-order, second-order, and third-order dynamic linear models. For comparison, static models are also developed. All the developed models are compared in terms of statistical fit and predictive performance. Based on the best fitted dynamic model, cycle-level crash probability and return level are calculated to measure signalized intersection safety at cycle level. The results show that dynamic models considerably outperform static models in terms of statistical fit and predictive performance. Further, the third-order dynamic model has the best performance, which is probably due to that the model incorporates two linear trends to respectively describe the variation of the coefficients as well as its change to better account for the variation in the effect of time-varying covariates. However, it should be noted that the third-order dynamic model development needs more computation time than other dynamic models, which may limit the application of the model.