Generalized Pareto Distribution Research Articles

The GARCH-EVT-Copula Approach to Investigating Dependence and Quantifying Risk in a Portfolio of Bitcoin and the South African Rand

This study uses a hybrid model of the exponential generalised auto-regressive conditional heteroscedasticity (eGARCH)-extreme value theory (EVT)-Gumbel copula model to investigate the dependence structure between Bitcoin and the South African Rand, and quantify the portfolio risk of an equally weighted portfolio. The Gumbel copula, an extreme value copula, is preferred due to its versatile ability to capture various tail dependence structures. To model marginals, firstly, the eGARCH(1, 1) model is fitted to the growth rate data. Secondly, a mixture model featuring the generalised Pareto distribution (GPD) and the Gaussian kernel is fitted to the standardised residuals from an eGARCH(1, 1) model. The GPD is fitted to the tails while the Gaussian kernel is used in the central parts of the data set. The Gumbel copula parameter is estimated to be α=1.007, implying that the two currencies are independent. At 90%, 95%, and 99% levels of confidence, the portfolio’s diversification effects (DE) quantities using value at risk (VaR) and expected shortfall (ES) show that there is evidence of a reduction in losses (diversification benefits) in the portfolio compared to the risk of the simple sum of single assets. These results can be used by fund managers, risk practitioners, and investors to decide on diversification strategies that reduce their risk exposure.

Wave Height Estimation with a Short Return Period

Bisection and grid search methods are proposed as wave height estimation techniques for short return periods (hereafter RP; reference, one year) using long-term wave monitoring data. The proposed method is compared and evaluated with the estimation results using GP (generalized Pareto) and GEV (generalized extreme value) distribution functions, which are widely used as extreme value analysis methods. The wave height data used for the estimation were KMA Ulleungdo Ocean Buoy’s wave height data observed for 12 years from 2012 to 2023. The estimation results show that the annual frequency wave height is 4.55 m, the 90% confidence interval (±5%) is [4.18, 4.69] (m), and the confidence interval for the RP is [0.58, 1.42] (years). The difference from the GP and GEV estimation results (4.61 m and 4.53 m, respectively) was statistically ‘no significance difference.’ The method proposed in this study can estimate design variables for RPs without assumptions on candidate extreme distribution functions or parameter estimation procedures, so it can be used to estimate wave heights for short RPs of one year or less when observation data of approximately ten years or more are available.

Strong Linkage Between Observed Daily Precipitation Extremes and Anthropogenic Emissions Across the Contiguous United States

AbstractThe results of probabilistic event attribution studies depend on the choice of the extreme value statistics used in the analysis, particularly with the arbitrariness in the selection of appropriate thresholds to define extremes. We bypass this issue by using the Extended Generalized Pareto Distribution (ExtGPD), which jointly models low precipitation with a generalized Pareto distribution and extremes with a different Pareto tail, to conduct daily precipitation attribution across the contiguous United States (CONUS). We apply the ExtGPD to 12 general circulation models from the Coupled Model Intercomparison Project Phase 6 and compare counterfactual scenarios with and without anthropogenic emissions. Observed precipitation by the Climate Prediction Center is used for evaluating the GCMs. We find that greenhouse gases rather than natural variability can explain the observed magnitude of extreme daily precipitation, especially in the temperate regions. Our results highlight an unambiguous linkage of anthropogenic emissions to daily precipitation extremes across CONUS.

Multivariate generalized Pareto distributions along extreme directions

Multivariate generalized Pareto distributions along extreme directions

A new Bayesian method for estimation of value at risk and conditional value at risk

AbstractValue at Risk (VaR) and Conditional Value at Risk (CVaR) have become the most popular measures of market risk in Financial and Insurance fields. However, the estimation of both risk measures is challenging, because it requires the knowledge of the tail of the distribution. Therefore, Extreme Value Theory initially seemed to be one of the best tools for this kind of problems, because using peaks-over-threshold method, we can assume the tail data approximately follow a Generalized Pareto distribution (GPD). The main objection to its use is that it only employs observations over the threshold, which are usually scarce. With the aim of improving the inference process, we propose a new Bayesian method that computes estimates built with all the information available. Informative prior Bayesian (IPB) method employs the existing relations between the parameters of the loss distribution and the parameters of the GPD that models the tail data to define informative priors in order to perform Metropolis–Hastings algorithm. We show how to apply IPB when the distribution of the observations is Exponential, stable or Gamma, to make inference and predictions. .Afterwards, we perform a thorough simulation study to compare the accuracy and precision of the estimates computed by IPB and the most employed methods to estimate VaR and CVaR. Results show that IPB provides the most accurate, precise and least biased estimates, especially when there are very few tail data. Finally, data from two real examples are analysed to show the practical application of the method.

Research on extreme value estimations and dynamic coefficients of wheel load for high-speed railway based on generalized Pareto distribution theory

The design value of wheel load is one of the key design parameters for high-speed railway ballastless track structure, which essentially belongs to the extreme estimation of wheel load. Based on the generalized Pareto distribution theory, this article carries on the research on the extreme value estimations of wheel load excited by whole wavelength range irregularity, which provides a basis for the design of ballastless track structure. When using automatic threshold selection method, sample declustering can improve the independence between data. The shape parameter screening method is proposed for the extreme value estimations of wheel load, and the corresponding sample size and shape parameter range for each cluster are determined. A reasonable sample size is determined, and the mean of the extreme estimations of wheel load is used as the final extreme estimation of wheel load. The dynamic coefficients that match the design life of the ballastless track structure are obtained.

Prediction intervals for future Pareto record claims

AbstractStochastic models and methods for quantifying extreme events are of interest in numerous disciplines. Within this paper, statistical prediction of extreme claims or losses in insurance industry is considered based on upper record values, which describe successively largest observations in a sequence of data over time. The problem of predicting a future record value (here in particular, a future record claim) based on a sequence of previously observed record values (here, past record claims) is addressed by means of prediction intervals. For an underlying Pareto distribution, respective exact and approximate intervals from the literature are summarized and modified and new ones are developed. In a simulation study, these prediction intervals are evaluated and compared regarding coverage frequency and length. The impact of the number of observed record values as well as the choice of the Pareto distribution is discussed. In the case of a small number of record values, the use of k-th record values is considered as an option for statistical analyses to predict, e.g., second largest record claims. Selected prediction methods are applied to several real data sets, which turn out to perform well and to be able to capture the magnitude of future record claims, even for fairly small numbers of record observations. For comparison, generalized Pareto distributions are fitted to real data sets and a corresponding point predictor as well as a respective upper prediction interval for the next record to appear are derived and evaluated.

Air Pollution Forecasting in a Regional Context for Sustainable Management

The aim of this research was to develop and apply a statistical model that can be used to forecast long-term daily maximum particulate matter with a diameter of less than 2.5 microns (PM2.5) concentrations. In order to predict the daily maximum PM2.5 concentrations in the northeastern region of Thailand, the extreme value theory was analyzed, and an appropriate distribution model was identified by employing the Generalized Pareto distribution (GPD). The data of daily maximum PM2.5 concentrations during the years 2021–2023 obtained from six stations was used. These stations are located in Khon Kaen, Loei, Nakhon Ratchasima, Nong Khai, Nakhon Phanom, and Ubon Ratchathani provinces. The results of this study reveal that the GPD is appropriate based on the results of Kolmogorov-Smirnov Statistics Test. Estimating the return levels during the following return periods: 2 years, 5 years, 10 years, 25 years, 50 years, and 100 years showed that the area in the upper northeastern region, particularly Loei and Nakhon Phanom, has daily maximum PM2.5 concentrations above 500 micrograms per cubic meter. These results can also be used as information to support decision-making when conducting response planning in high-risk areas, which can be helpful for efficient resource planning and prevention actions. Doi: 10.28991/ESJ-2024-08-05-024 Full Text: PDF

Predicting storm surge extremes on the Southeast Brazilian Coast: Long-term projections with neural networks

Extreme sea level events, resulting from the confluence of tides and storm surges, pose a significant threat to coastal populations and economies. The escalating risks associated with these events are exacerbated by climate change, manifesting in heightened storm intensity, increased frequency, and rising sea levels. Precise estimation of the probability of extreme storm surges is crucial for effective coastal management and adaptation. However, utilizing historical storm data is challenging due to data scarcity and the imperative to consider potential non-stationarity induced by climate change in predicting such events. This study addresses these challenges by introducing two neural network-based machine learning systems: a Multilayer Perceptron (MLP) and a Long Short-Term Memory (LSTM). Leveraging local and remote atmospheric and oceanic conditions, these systems project storm surges until 2060, incorporating climate projections. Trained and evaluated using sea level data from Imbetiba Port in Macaé, Rio de Janeiro, Brazil, the models utilize dynamic regionalization data from the RegCM4 and WW3 models, forced by HadGEM2-ES and MPI climate models. Both neural network models exhibited similar performance patterns, demonstrating high agreement in predicting storm surge heights with a 100-year return value, based on Imbetiba Port data. Projections utilized peaks-over-threshold (POT) methods, and extremes were calculated using a generalized Pareto distribution (GPD). Long-term projections indicated a 28 % increase (MLP ANN) and a substantial 70 % increase (LSTM RNN) in estimating extreme values, surpassing the observed storm surge of 0.67 m. Projected mean values were 0.86 m for the MLP network and 1.15 m for the LSTM network, providing valuable insights into the potential amplification of extreme sea level events in the studied region.

Extrapolation Framework and Characteristic Analysis of Load Spectrum for Agriculture General Power Machinery

As a crucial step in food production, tillage and land preparation play a pivotal role in achieving sustainable crop production and improving the soil environment. However, accurate assessment of the load that agricultural machinery implements during the operation process has always been a vexing problem that needs urgent solutions. In this paper, an extrapolation and reconstruction framework for the time-domain load is constructed based on the probability-weighted moments (PWM) estimation and the peaks-over-threshold function, and the load spectrum is obtained for agriculture general power machinery. Firstly, the load acquisition system was developed, the traction resistance and output torque of the tractor were measured, and the collected load signals were preprocessed. Next, the mean excess function and PWM estimation are introduced to select the optimal threshold and generalized Pareto distribution (GPD) fitting parameters and the extreme load distribution that exceeds the threshold range is fitted. The extreme points in the original data are replaced by generating new extreme points that follow the GPD distribution, and the extrapolation of the load spectrum is achieved. Finally, the real extrapolated load spectrum was validated based on statistical characteristics and rainflow counting analysis, and the correlation coefficient between the fitting data and the extreme load samples was greater than 0.99. It can retain the load sequence characteristics of the original load to a great extent, truly reflecting the load state during the operation of agricultural machinery. Meanwhile, the characteristics of the load spectrum can be accurately obtained, such as extreme, mean, and amplitude values, and the real load during deep loosening and rotary tillage are accurately described. The values provide more authentic and reliable data support for the subsequent selection of optimal operating parameters, reliability design of the power transmission system, and the life assessment of the agricultural implements.

Estimation and Bayesian Prediction of the Generalized Pareto Distribution in the Context of a Progressive Type-II Censoring Scheme

The generalized Pareto distribution plays a significant role in reliability research. This study concentrates on the statistical inference of the generalized Pareto distribution utilizing progressively Type-II censored data. Estimations are performed using maximum likelihood estimation through the expectation–maximization approach. Confidence intervals are derived using the asymptotic confidence intervals. Bayesian estimations are conducted using the Tierney and Kadane method alongside the Metropolis–Hastings algorithm, and the highest posterior density credible interval estimation is accomplished. Furthermore, Bayesian predictive intervals and future sample estimations are explored. To illustrate these inference techniques, a simulation and practical example are presented for analysis.

Extreme particulate matter exposure at traffic intersections in a densely populated city

Commuters are exposed to significantly high pollution levels at traffic intersections. This study utilized the generalized Pareto distribution to model commuters’ exposure to extreme particulate matter (PM) levels across 36 traffic intersections in Varanasi, India. A Bayesian hierarchical framework was deployed to account for the seasonal variation. The monthly return levels for extreme PM2.5 (PM10) were 589 (1127), 474 (961), and 429 (902) µg/m3 during winter, spring, and summer, respectively. The extreme PM2.5 and PM10 concentrations exceeded the NAAQS severe category for all three seasons. There is a 0.72 % (1.47 %) chance that during winter, PM2.5 (PM10) levels would exceed that of the Delhi smog event (PM2.5: 585 µg/m3, PM10: 989 µg/m3). These findings raise concerns about public health and the environment, particularly in winter. The results would guide policymakers in enforcing stringent measures to reduce extreme exposures at traffic intersections in densely populated cities.

The adaptive mean estimation of heavy tailed distribution under random censoring

We propose an alternative estimator of the heavy tailed mean under censored data relies with the threshold t which divides it into two parts for x ≤ t and x > t respectively. In the first parts we present an empirical distribution function under random censoring (rc). Then, we check the convergence of Lyapunov condition to ensuring the asymptotic normality of this parts. For the second parts we use the Peaks-Over-Threshold (POT) method of the tail distribution and fitting to the generalized Pareto distribution (GPD). Intentionally to give an alternative estimator of the tail expected value for the heavy tailed distribution when x > t under random censoring relies the KIB estimators of the GPD parameters. Ditto, we ensuring the asymptotic normality property of the KIB estimators with bias reduced and we present the asymptotic normality of the second parts when x > t. An asymptotically normally distributed estimate for the heavy tailed mean estimation with infinite variance under censored data is introduced by combining the two parts. Finally, numerical examples are presented at the end of the paper to demonstrate the reliability of this estimation procedure and to better illustrate the results of this paper.

Impacts of climate and land-use change on flood events with different return periods in a mountainous watershed of North China

Study regionLuanhe River Basin in North China Study focusGlobal warming exacerbates hydrological cycles, leading to increased intensity and frequency of floods. Regional studies on the impacts of climate change and land cover changes on flood events in recent decades are particularly important for flood prevention and risk assessment. In this study, the Peak Over Threshold- Generalized Pareto Distribution (POT-GPD) method is used to fit extreme runoff values in the Luanhe River Basin from 1961 to 2018 to identify floods with different return periods. Six flood behavior indicators are adopted to analyze the flood characteristics and extreme climate indices are also used for breakpoint testing and trend analysis. Various scenarios are established as the input of the SWAT model to study the impacts of climate change and land-use change on the flood events. New hydrological insights for the regionThe results indicate that the 2-year and 20-year return period floods are primarily rapid-rise single-peak floods, dominated by short-duration intense precipitation. The 5-year and 10-year return period floods are mainly multi-peak floods, primarily caused by long-duration precipitation. Floods in the Luanhe River Basin are primarily influenced by climate change, however, flood events do not directly correspond to the return periods of precipitation events. Forest is the most effective land-use type for flood retention. It could reduce flood magnitudes, especially for the 20-year return period floods. However, increasing forest cover does not effectively reduce the occurrence of the 5-year or 10-year floods, i.e. multi-peak floods. With more extreme rainfall events triggered by future climate change, particular attention should be paid to long-duration precipitation events, which may result in more severe disasters and economic losses.

How to define the threshold of takeover response ability of different drivers in conditional automated driving

In conditional automated driving, the takeover response ability threshold is necessary for driver qualification assessment and liability division of automated vehicle accidents. The primary objective of this study is to establish a clear and quantifiable threshold for drivers’ takeover response ability in conditional automated driving scenarios. This threshold aims to serve as a benchmark for evaluating drivers’ readiness and developing safety regulations in automated driving. We designed 18 takeover events and invited 42 drivers to participate in the driving simulation experiment, and obtained their takeover time data. First, we analyze the differences of takeover time among drivers with different attributes (gender, age, driving year). Second, based on the Peaks Over Threshold and the generalized Pareto distribution model, we use the graphic method to calculate the range of takeover time threshold for drivers with different attributes. The result shows that the difference in the threshold range of takeover time between male and female drivers is relatively tiny. There are differences in the threshold range of takeover time for different age drivers, and the threshold is negatively correlated with age. Drivers with high driving experience within a safe range are allowed to have longer takeover times. Finally, the rationality of the takeover time threshold for drivers with different attributes has been verified. The return level curves are approximately linear (R2 > 0.77), indicating that the GPD model can capture the overall trend of the return level, which is changing with the probability level. This proves that the takeover time threshold is reasonable. This study uses TTCmin to calibrate takeover safety, and the takeover time threshold has a good classification performance for takeover safety (accuracy > 85 %). The above content proves the rationality of the takeover time threshold. The contribution of this study is to calculate the takeover time threshold of drivers with different attributes, which can help regulatory authorities assess the driver’s takeover response ability and support the liability division of automated vehicle accidents.

Calculating Premiums for Extreme-Tail Risks with Liability Claims and Nuclear Verdicts

Purpose: When insurance claims, particularly liability claims and nuclear verdicts, are governed by fat-tailed distributions, considerable uncertainty exists about the value of the tail index. Material and Methods: Using the theory of risk aversion, this paper establishes a new premium principle (the power principle – analogous to the exponential principle for thin-tailed claims) and investigates its properties. Findings: Applied to claims arising from generalized Pareto distributions, the resultant premium is shown to be the ratio of the two largest expected claims. This structure provides a natural model for incorporating tail-index uncertainty into premiums. The theory is illustrated through possible ‘premiums’ for liability claims and nuclear verdicts, utilizing the consistent pattern of observed extremes. Implications to Theory, Practice and Policy: By integrating statistical methods for tail index estimation and addressing the inherent uncertainty, the power principle offers a robust framework for determining premiums in high-risk environments. The paper concludes with practical implications for insurers, highlighting the need for advanced risk management strategies and regulatory considerations in dealing with extreme liability claims.

Continental Scale Regional Flood Frequency Analysis: Combining Enhanced Datasets and a Bayesian Framework

Flood frequency analysis at large scales, essential for the development of flood risk maps, is hindered by the scarcity of gauge flow data. Suitable methods are thus required to predict flooding in ungauged basins, a notoriously complex problem in hydrology. We develop a Bayesian hierarchical model (BHM) based on the generalized extreme value (GEV) and the generalized Pareto distribution for regional flood frequency analysis at high resolution across a large part of North America. Our model leverages annual maximum flow data from ≈20,000 gauged stations and a dataset of 130 static catchment-specific covariates to predict extreme flows at all catchments over the continent as well as their associated statistical uncertainty. Additionally, a modification is made to the data layer of the BHM to include peaks over threshold flow data when available, which improves the precision of the discharge level estimates. We validated the model using a hold-out approach and found that its predictive power is very good for the GEV distribution location and scale parameters and improvable for the shape parameter, which is notoriously hard to estimate. The resulting discharge return levels yield a satisfying agreement when compared with the available design peak discharge from various government sources. The assessment of the covariates’ contributions to the model is also informative with regard to the most relevant underlying factors influencing flood-inducing peak flows. According to the developed aggregate importance score, the key covariates in our model are temperature-related bioindicators, the catchment drainage area and the geographical location.

Anthropogenic and meteorological effects on the counts and sizes of moderate and extreme wildfires

AbstractThe growing frequency and size of wildfires across the US necessitates accurate quantitative assessment of evolving wildfire behavior to predict risk from future extreme wildfires. We build a joint model of wildfire counts and burned areas, regressing key model parameters on climate and demographic covariates. We use extended generalized Pareto distributions to model the full distribution of burned areas, capturing both moderate and extreme sizes, while leveraging extreme value theory to focus particularly on the right tail. We model wildfire counts with a zero‐inflated negative binomial model, and join the wildfire counts and burned areas sub‐models using a temporally‐varying shared random effect. Our model successfully captures the trends of wildfire counts and burned areas. By investigating the predictive power of different sets of covariates, we find that fire indices are better predictors of wildfire burned area behavior than individual climate covariates, whereas climate covariates are influential drivers of wildfire occurrence behavior.

The SPAR Model: A New Paradigm for Multivariate Extremes: Application to Joint Distributions of Metocean Variables

Abstract This paper presents the application of a new multivariate extreme value model for the estimation of metocean variables. The model requires fewer assumptions about the forms of the marginal distributions and dependence structure compared to existing approaches, and provides a flexible and rigorous framework for modeling multivariate extremes. The method involves a transformation of variables to polar coordinates. The tail of the radial variable is then modeled using the generalized Pareto distribution, with parameters conditional on angle, providing a natural extension of univariate theory to multivariate problems. The resulting model is referred to as the semi-parametric angular-radial (SPAR) model. We consider the estimation of the joint distributions of (1) wave height and wave period, and (2) wave height and wind speed. We show that the SPAR model provides a good fit to the observations in terms of both the marginal distributions and dependence structures. The use of the SPAR model for estimating long-term extreme responses of offshore structures is discussed, using some simple response functions for floating structures and an offshore wind turbine with monopile foundation. We show that the SPAR model is able to accurately reproduce response distributions, and provides a realistic quantification of uncertainty.

Degree distributions in networks: Beyond the power law

AbstractThe power law is useful in describing count phenomena such as network degrees and word frequencies. With a single parameter, it captures the main feature that the frequencies are linear on the log‐log scale. Nevertheless, there have been criticisms of the power law, for example, that a threshold needs to be preselected without its uncertainty quantified, that the power law is simply inadequate, and that subsequent hypothesis tests are required to determine whether the data could have come from the power law. We propose a modeling framework that combines two different generalizations of the power law, namely the generalized Pareto distribution and the Zipf‐polylog distribution, to resolve these issues. The proposed mixture distributions are shown to fit the data well and quantify the threshold uncertainty in a natural way. A model selection step embedded in the Bayesian inference algorithm further answers the question whether the power law is adequate.