Generalized Pareto Distribution Research Articles

Applying the Generalized Pareto Principle to Predict Peak Temperatures in Northeast Algeria

This study employs the Generalized Pareto Distribution (GPD) to model high temperature events at the Batna station, strategically applying varying thresholds to enhance accuracy and mitigate tail distribution bias. This was done by analyzing historical rainfall data from 1981 to 202. Using maximum likelihood estimation and an innovative approach to fitting multi-threshold GPDs, stable thresholds were established, particularly identifying the Pareto II type as the most suitable model for temperatures exceeding 25°C. Analysis revealed a consistent average temperature of 27.52°C, with a negative skewness indicating a bias toward lower values within extreme temperatures. The study also predicts return levels for different periods, providing critical temperature thresholds expected to be exceeded every 2, 20, 50, and 100 years. These findings contribute to understanding extreme temperature dynamics in the region, supporting urban planning, infrastructure design, and climate resilience strategies.

Semi-Empirical Approach to Evaluating Model Fit for Sea Clutter Returns: Focusing on Future Measurements in the Adriatic Sea.

A method for evaluating Kullback-Leibler (KL) divergence and Squared Hellinger (SH) distance between empirical data and a model distribution is proposed. This method exclusively utilises the empirical Cumulative Distribution Function (CDF) of the data and the CDF of the model, avoiding data processing such as histogram binning. The proposed method converges almost surely, with the proof based on the use of exponentially distributed waiting times. An example demonstrates convergence of the KL divergence and SH distance to their true values when utilising the Generalised Pareto (GP) distribution as empirical data and the K distribution as the model. Another example illustrates the goodness of fit of these (GP and K-distribution) models to real sea clutter data from the widely used Intelligent PIxel processing X-band (IPIX) measurements. The proposed method can be applied to assess the goodness of fit of various models (not limited to GP or K distribution) to clutter measurement data such as those from the Adriatic Sea. Distinctive features of this small and immature sea, like the presence of over 1300 islands that affect local wind and wave patterns, are likely to result in an amplitude distribution of sea clutter returns that differs from predictions of models designed for oceans or open seas. However, to the author's knowledge, no data on this specific topic are currently available in the open literature, and such measurements have yet to be conducted.

Topp-Leone generalization of the Generalized Pareto distribution and its impact on Extreme value modelling

Extreme Value theory (EVT) is a phenomenon used to model rare or extreme events and has been useful in well-known areas such as finance, economics, hydrology, insurance, etc. In this paper, we combine EVT and Bayesian statistics to estimate the extreme value index and other distribution parameters. EVT studies the behavior of the tails of the distribution, while Bayesian statistics allows us to incorporate prior knowledge of the parameters. The interdependence between these two statistical branches allows us to account for uncertainty in parameter and tail estimation. Block maxima and Peaks over Threshold are EVT divisions that are used to model observations. In this paper we use the Peaks over Threshold approach. The generalized Pareto distribution is a Peaks over Threshold distribution. Existing literature studied the generalizations and extensions of the generalized Pareto distribution. These extensions mostly focus on the positive domain of attraction. In this paper we contribute to the study of EVT by considering both the negative and positive domains of attraction. We consider the (Topp and Leone, 1955) generalization for the generalized Pareto distribution. We show, by means of a simulation study, that this distribution can effectively estimate the extreme value index and that it is less sensitive to threshold selection than the normal generalized Pareto distribution.

A calibration method for projecting future extremes via a linear mapping of parameters

AbstractIn order to study potential impacts arising from climate change, future projections of numerical model output often must be calibrated to be comparable to observations. Rather than calibrating the data values themselves, we propose a novel statistical calibration method for extremes that assumes there exists a linear relationship between parameters associated with model output and parameters associated with observations. This approach allows us to capture uncertainty in both parameter estimates and the linear calibration, which we achieve via bootstrap. To focus on extreme behavior, we assume both model output and observations have distributions composed of a mixture model combining a Weibull distribution with a generalized Pareto distribution for the tail. A simulation study shows good coverage rates. We apply the method to project future daily-averaged river runoff at the Purgatoire River in southeastern Colorado.

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.

Time-domain extrapolation of vehicle durability load spectrum based on load decomposition and POT model threshold preference

In vehicle structural fatigue life assessment, the time-domain Peak over Threshold (POT) extrapolation method is widely used due to its ability to overcome testing constraints and accurately extrapolate whole life-cycle load while maintaining realistic probability distribution. However, random road load spectra often contain trend components influenced by terrain and driving maneuvers, which may reduce the POT method’s adaptability and lead to extrapolation distortions. Existing POT extrapolation methods also struggle with reliance on single threshold criteria and inadequate extreme value fitting. To address these issues, this paper proposes an improved time-domain POT extrapolation method that incorporates load decomposition and threshold preference. The wavelet decomposition is utilized to extract the principal and trend components from random signals, allowing for targeted extrapolation of principal component load that embodies the main characteristics. Meanwhile, during the extrapolation process, a comprehensive threshold preference model utilizing multiple evaluation criteria for Generalized Pareto Distribution (GPD) fitting is established, with the genetic algorithm determining the optimal threshold. Then, new extreme values are generated through GPD parameter estimation and Monte-Carlo simulation to replace the original ones. The extrapolated principal component load is subsequently superimposed with the original trend load to complete the extrapolation. Validation with a prototype vehicle’s durability spectrum under real driving conditions demonstrates the method’s efficacy in preventing distortions and achieving accurate threshold determination, with predicted load characteristics and fatigue damage highly aligning with measured verification data. It can provide a valuable technical reference for whole life-cycle fatigue load acquisition of mechanical structures.

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

Statistical analysis of global ocean significant wave heights from satellite altimetry over the past 2 decades

Abstract. The analysis of global ocean surface waves and of long-term changes requires accurate time series of waves over several decades. Such time series have previously only been available from model reanalyses or from in situ observations. Now, altimetry provides a long series of observations of significant wave heights (SWHs) in the global ocean. The aim of this study is to analyse the climatology of significant wave heights and extreme significant wave heights derived from remote sensing in the global ocean and their long-term trends from 2002 to 2020 using different statistical approaches as the mean, the 95th percentile, and the 100-year return level of SWH. The mean SWH and the 95th percentile of SWH are calculated for two seasons: January, February, and March as well as July, August, and September and for each year. A trend is then estimated using linear regression for each cell in the overall grid. The 100-year return levels are determined by fitting a generalized Pareto distribution to all exceedances over a high threshold. The trend in 100-year return level is estimated using the transformed stationary approach, which, to our knowledge, is used for the first time to draw a global map based on altimetry. Predominantly large positive trends over 2002–2020 for both SWH and extreme SWH are mostly found in the Southern Hemisphere, including the South Atlantic, the Southern Ocean, and the southern Indian Ocean, which is consistent with previous studies. In the North Atlantic, SWH has increased poleward of 45° N, corroborating what was concluded in the Fifth IPCC Assessment Report; however, SWH has also largely decreased equatorward of 45° N in wintertime. The 100-year return levels of SWH have significantly increased in the North Atlantic and in the eastern tropical Pacific, where the cyclone tracks are located. Finally, in this study we find trends of SWH and 95th percentile of SWH over 2002–2020 to be much higher than those indicated in the literature for the period 1985–2018.

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.

Structural Properties of the Alpha Power Exponentiated Generalized Pareto Distribution with Applications

Structural Properties of the Alpha Power Exponentiated Generalized Pareto Distribution with Applications