Generalized Pareto Distribution Parameters Research Articles

Statistical Inference and Application of Asymmetrical Generalized Pareto Distribution Based on Peaks-Over-Threshold Model

Generalized Pareto distribution (GPD), an asymmetrical distribution, primarily models exceedances over a high threshold in many applications. Within the peaks-over-threshold (POT) framework, we consider a new GPD parameter estimation method to estimate a common tail risk measure, the value at risk (VaR). The proposed method is more suitable for the POT framework and makes full use of data information. Specifically, our estimation method builds upon the generalized probability weighted moments method and integrates it with the nonlinear weighted least squares method. We use exceedances for the GPD, minimizing the sum of squared differences between the sample and population moments of a function of GPD random variables. At the same time, the proposed estimator uses three iterations and assigns weight to further improving the estimated performance. Under Monte Carlo simulations and with a real heavy-tailed dataset, the simulation results show the advantage of the newly proposed estimator, particularly when VaRs are at high confidence levels. In addition, by simulating other heavy-tailed distributions, our method still exhibits good performance in estimating misjudgment distributions.

Fitting generalized Pareto distribution based on the expected order statistics and its application for ice accretion hazard mapping

The extreme value analysis of the ice accretion thickness can be carried out based on the peaks-over-threshold (POT) approach to take advantage of the available rare samples in assessing the ice accretion thickness hazard. For the POT approach, the generalized Pareto distribution (GPD) with location, scale, and shape parameters has been frequently employed, although the estimators of parameters of GPD by using existing fitting methods could be associated with significant bias or variability. In the present study, we propose a new nonlinear least-squares fitting method by considering the expected order statistics. We compare its performance with some existing methods through simulation analysis, showing that the proposed method outperforms some well-known existing methods for a range of values of the shape parameter.We then apply the POT approach with GPD as well as the block maximum approach with Gumbel distribution to estimate T-year return period value of the annual maximum ice accretion thickness, xT, for Canadian sites. The analysis results indicate that xT is sensitive to whether the POT or the annual maximum approach with Gumbel distribution is employed. For the considered cases, the use of GPD fitted to POT values is preferred. Also, based on the wind-on-ice analysis results, it is suggested a companion wind load factor of about 0.5 to 0.55 could be adopted.

Parameter Estimation for Sea Clutter Pareto Distribution Model Based on Variable Interval

The generalized Pareto (GP) distribution model is often used to describe the amplitude statistical feature of sea clutter. Generally, the parameters of GP distribution are estimated by moments estimators. However, when the sea state is high, the appearance of sea spikes will increase the echo of the anomalous scattering units, which leads to a decrease in the parameter estimation accuracy and target detection performance. To improve the parameter estimation accuracy, this paper proposes a novel parameter estimation method based on variable intervals. Considering the local properties of sea clutter, we take a variable interval of the entire sea clutter series for parameter estimation, where the interval position is selected according to the sea state conditions. For contrast, the bipercentile parameter estimation and truncate moment estimation are also introduced. Finally, the experiment based on the real measured X-band sea clutter datasets indicates that the proposed method outperforms the state-of-the-art moments estimators.

Baseline Methods for the Parameter Estimation of the Generalized Pareto Distribution.

In the parameter estimation of limit extreme value distributions, most employed methods only use some of the available data. Using the peaks-over-threshold method for Generalized Pareto Distribution (GPD), only the observations above a certain threshold are considered; therefore, a big amount of information is wasted. The aim of this work is to make the most of the information provided by the observations in order to improve the accuracy of Bayesian parameter estimation. We present two new Bayesian methods to estimate the parameters of the GPD, taking into account the whole data set from the baseline distribution and the existing relations between the baseline and the limit GPD parameters in order to define highly informative priors. We make a comparison between the Bayesian Metropolis–Hastings algorithm with data over the threshold and the new methods when the baseline distribution is a stable distribution, whose properties assure we can reduce the problem to study standard distributions and also allow us to propose new estimators for the parameters of the tail distribution. Specifically, three cases of stable distributions were considered: Normal, Lévy and Cauchy distributions, as main examples of the different behaviors of the tails of a distribution. Nevertheless, the methods would be applicable to many other baseline distributions through finding relations between baseline and GPD parameters via studies of simulations. To illustrate this situation, we study the application of the methods with real data of air pollution in Badajoz (Spain), whose baseline distribution fits a Gamma, and show that the baseline methods improve estimates compared to the Bayesian Metropolis–Hastings algorithm.

Characterizing Wind Power Forecast Error Using Extreme Value Theory and Copulas

Wind energy is one of the fastest-growing renewable energy sources in the world. However, wind power is variable in all timescales. This variability is difficult to predict with perfect certainty, with potentially significant financial implications when rare extreme forecast errors occur. This paper focuses on three key aspects associated with the extreme errors of geographically distributed wind farms: suitable parametric distribution representation, effects of diurnality, seasonality and larger atmospheric circulations, and modeling multivariate distribution. The paper shows that some of the distributions commonly used for modeling forecast errors may be inappropriate in representing extreme errors. As the first contribution, this paper fits a Generalized Pareto distribution (GPD) from extreme value theory to achieve a better estimation of extreme errors. In the second contribution, this paper splits extreme errors by hour, month, and atmospheric states to investigate the statistical regularities of GPD parameters along diurnal and seasonal timescales and larger atmospheric circulations. In the third contribution, this paper uses copula functions to model multivariate extreme error distribution and investigates their effectiveness in providing a regional view of extreme errors. This paper tests the proposed methodology using the forecast error data obtained from 29 wind farms in South Africa. The results show that GPD outperforms commonly used distributions. Extreme errors have strong diurnal and seasonal components and vary significantly between SOM nodes. Copulas can be useful in providing a regional view of extreme errors. This paper improves the estimation of extreme errors, which is an important step toward better operating reserve allocation.

Distributional Regression Forests Approach to Regional Frequency Analysis With Partial Duration Series

AbstractRegional flood frequency analysis (RFFA) is widely used to quantify flood risk at ungauged and sparsely gauged locations. There are minimal attempts to use partial duration series (PDS) for RFFA, though the use of PDS instead of widely used annual maximum series (AMS) can offer some advantages. This article contributes two novel random/regression forests (RFs)‐based methodologies, namely generalized pareto distribution (GPD)‐based distributional RFs (DRFs) and multivariate RFs (MVRFs), for RFFA with PDS. The RFs facilitate modeling interactions between predictors and their complex relationships with the predictands without explicitly specifying them. The DRFs and MVRFs comprise an ensemble of corresponding regression trees, each constructed by recursive binary partitioning of the feature space into meaningful segments. The proposed DRFs account for the sampling uncertainty of PDS in the partitioning and parameter estimation. In DRFs (MVRFs), quantile estimates for an ungauged site are obtained using maximum likelihood estimates (expected values) of GPD parameters corresponding to the segments to which the site belongs. The potential of DRFs and MVRFs relative to two recently proposed techniques (univariate RFs‐based quantile regression, generalized additive model based on GPD) is demonstrated through Monte‐Carlo simulation experiments and a study on 1,031 watersheds in the United States. The key features influencing scale and shape parameters of GPD fitted to PDS of the watersheds are identified as drainage area and 24‐hr rainfall intensity corresponding to 2‐year return period, respectively. Those identified for shape parameter differ from key features known based on analysis with AMS and generalized extreme value distribution.

Modeling of Extreme Crude Oil Price using the Generalized Pareto Distribution: Brent and West Texas Benchmark Price

Background and objective: Crude oil is an essential commodity in many countries of the world. This work studies the risk involved in the extreme crude oil price, using the daily crude oil price of the Brent and the West Texas benchmark from year 1990 to 2019. Materials and methods: The Peak Over Threshold (POT) approach of the Generalized Pareto Distribution (GPD) was used to model the extreme crude oil price while the value at risk and the expected shortfall was used to quantify the risk involved in extreme price of crude oil. The GPD, using the Q-Q plot was found to be a good model for the extreme values of the crude oil price. Results: The Value at Risk (VaR) and the Expected Shortfall (ES) calculated at 90%, 95% and 99% with the Maximum Likelihood estimators of GPD parameters and the threshold values were found to decrease with increase in quantile for both benchmark. This shows that risk involved in extreme crude oil price will be borne only by the investors and public. Conclusion: It was also found that the VaR and ES of the Brent are higher than that of West Texas. This implies that it is safer to invest in West Texas crude oil.

Upper record values from the generalized Pareto distribution and associated statistical inference

We investigate point estimation and confidence interval estimation for the heavy-tailed generalized Pareto distribution (GPD) based on the upper record values. When the shape parameter is known, the bias-corrected moments estimators and maximum likelihood estimators (MLE) for the location and scale parameters are derived. However, in practice, the shape parameter is typically unknown. We propose the MLE by a new methodological approach for all three parameters of the heavy-tailed GPD when the shape parameter is unknown. Confidence intervals for the location and scale parameters are constructed by the equal probability density principle. If the shape parameter is known, we can find known distributions of the pivots of the location and scale parameters, but not approximate. While if the shape parameter is unknown, the distributions of the pivots are closed linked to an estimation of the shape parameter. The advantage of our method is that the proposed interval estimation provides the smallest confidence interval, regardless of whether the distribution of the pivot is symmetric. Extensive simulations are used to demonstrate the performance of the point estimation and confidence intervals estimation and show that our method outperforms the traditional technique in most cases.

A hierarchical bayesian peak over threshold approach for conflict-based before-after safety evaluation of leading pedestrian intervals

A hierarchical Bayesian peak over threshold (POT) approach is proposed for conflict-based before-after safety evaluation of Leading Pedestrian Intervals (LPI). The approach combines traffic conflicts of different sites and periods to develop a uniform generalized Pareto distribution (GPD) model for the treatment effect estimation. The hierarchical structure has three levels, a data level that consists of modeling the traffic conflict extremes through the POT approach, a latent process level that relates GPD parameters of the data level to certain covariates, and a prior level with prior distributions to characterize the latent process. The approach was applied to a before-after (BA) safety evaluation of leading pedestrian interval (LPI) in Vancouver, BC. Pedestrian-vehicle traffic conflicts were collected from treatment and control sites during the before and after periods using an automated computer vision analysis technique. The treatment effect was measured by the best fitted GPD model with the calculation of the odds ratio (OR). The overall treatment effect varies from 18.1%–20.9% in terms of reduction in the estimated extreme-serious conflicts. The treatment effect indicates a considerable improvement in pedestrian safety after the implementation of the LPI, and the consistent results demonstrate a reliable BA safety evaluation. As such, the proposed approach is recommended as a promising tool for BA safety studies, particularly in cases where collision data is limited.

An evaluation of automated GPD threshold selection methods for hydrological extremes across different scales

This study investigated core components of an extreme value methodology for the estimation of high-flow frequencies from agricultural surface water run-off. The Generalized Pareto distribution (GPD) was used to model excesses in time-series data that resulted from the ‘Peaks Over Threshold’ (POT) method. First, the performance of eight different GPD parameter estimators was evaluated through a Monte Carlo experiment. Second, building on the estimator comparison, two existing automated GPD threshold selection methods were evaluated against a proposed approach that automates the threshold stability plots. For this second experiment, methods were applied to discharge measured at a highly-instrumented agricultural research facility in the UK. By averaging fine-resolution 15-minute data to hourly, 6-hourly and daily scales, we were also able to determine the effect of scale on threshold selection, as well as the performance of each method. The results demonstrate the advantages of the proposed threshold selection method over two commonly applied methods, while at the same time providing useful insights into the effect of the choice of the scale of measurement on threshold selection. The results can be generalised to similar water monitoring schemes and are important for improved characterisations of flood events and the design of associated disaster management protocols.

Regression models for exceedance data: a new approach

The generalized Pareto distribution (GPD) is a family of continuous distributions used to model the tail of the distribution to values higher than a threshold u. Despite the advantages of the GPD representation, its shape and scale parameters do not correspond to the expected value, which complicates the interpretation of regression models specified using the GPD. This study proposes a linear regression model in which the response variable is a GPD, using a new parametrization that is indexed by mean and precision parameters. The main advantage of our new parametrization is the straightforward interpretation of the regression coefficients in terms of the expectation of the positive real line response variable, as is usual in the context of generalized linear models. Furthermore, we propose a model for extreme values, in which the GPD parameters (mean and precision) are defined on the basis of a dynamic linear regression model. The novelty of the study lies in the time variation of the mean and precision parameter of the resulting distribution. The parameter estimation of these new models is performed under the Bayesian paradigm. Simulations are conducted to analyze the performance of our proposed models. Finally, the models are applied to environmental datasets (temperature datasets), illustrating their capabilities in challenging cases in extreme value theory.

Application of the non-stationary peak-over-threshold methods for deriving rainfall extremes from temperature projections

Concerns about climate change are amplifying interest in future rainfall extremes. However, there are big differences between the statistics of rainfall extremes obtained using future rainfall time series produced from various climate models. Such large uncertainties create a lot of confusion in establishing climate change adaptation measures. Looking at future rainfall extremes at a particular site yields increasing trends in some climate models and decreasing trends in others. The spatial patterns of rate of change in rainfall extremes also vary widely, depending on the climate model. As a result, they often do not gain the public’s trust. We believe that this difficulty in obtaining credibility does not come from a lack of theory or technique, but from an approach that persuades the public of uncertain future rainfall extremes. In this study, we employed a novel approach to integrate a co-variate of the not-stationary Peak-Over-Threshold (POT) – Generalized Pareto distribution (GPD) model identified at each site with its future projection information for obtaining future rainfall extreme ensembles. Rainfall extremes are obtained from the observed rainfall time series using the POT method, and the scale parameter among GPD parameters are applied as a function of surface air temperature (SAT) or dew-point temperature (DPT). At this time, the threshold of the POT series is set to match the results of frequency analysis of the annual maximum series and the POT series for each site as much as possible. The behavior of future rainfall extremes is analyzed by inputting the future SAT or DAT produced from various climate models into the non-stationary frequency model using the co-variate. As a result of comparing the rainfall extremes obtained using the future rainfall time series directly with the future rainfall extremes obtained indirectly using the proposed method, it was found that the proposed approach projected future design rainfall depths with much less variation between climate models. The spatial pattern of rate of change was also consistent regardless of climate model. The proposed method is expected to contribute to the public’s confidence in future rainfall extremes under climate change scenarios and to be of practical help in formulating reasonable climate change adaptation policies.

Tree-based Threshold Model for Non-stationary Extremes with Application to the Air Pollution Index Data

Air pollution index (API) is a common tool used to describe the air quality in the environment. High level of API indicates the greater level of air pollution which will gives bad impact on human health. Statistical model for high level of API is important for the purpose of forecasting the level of API so that the public can be warned. In this study, extremes of API are modelled using Generalized Pareto Distribution (GPD). Since the values of API are determined by the value of five pollutants namely sulphur dioxide, nitrogen dioxide, carbon monoxide, ozone and suspended particulate matter, data on API exhibit non-stationarity. Standard method for modelling the non-stationary extremes using GPD is by fixing the high constant threshold and incorporating the covariate model in the GPD parameters for data above the threshold to account for the non-stationarity. However, high constant threshold value might be high enough on certain covariate for GPD approximation to be a valid model for extreme values, but not on the other covariates which leads to the violation of the asymptotic basis of GPD model. New method for the threshold selection in non-stationary extremes modelling using regression tree is proposed to the API data. Regression tree is used to partition the API data into a stationary group with similar covariate condition. Then, a high threshold value can be applied within a group. Study shows that model for extremes of API using tree-based threshold gives a good fit and provides an alternative to the model based on standard method.

Confidence Intervals of the Generalized Pareto Distribution Parameters Based on Upper Record Values

In this paper, we proposed a new efficient approach to construct confidence intervals for the location and scale parameters of the generalized Pareto distribution (GPD) when the shape parameter is known. The superiority of our method is that the distributions of pivots are exact, but not approximate distributions. The proposed interval estimation provides the shortest interval for the GPD parameter whether or not the confident distribution of the pivot is symmetric. We first estimate the location and scale parameters of the GPD using least squares and then, construct confidence intervals based on the equal probability density principle. The results of various simulation studies illustrate that our interval estimators show the better performance than competing method.

Generalized Additive Models for Exceedances of High Thresholds With an Application to Return Level Estimation for U.S. Wind Gusts

ABSTRACT Generalized additive model (GAM) forms offer a flexible approach to capturing marginal variation. Such forms are used here to represent distributional variation in extreme values and presented in terms of spatio-temporal variation, which is often evident in environmental processes. A two-stage procedure is proposed that identifies extreme values as exceedances of a high threshold, which is defined as a fixed quantile and estimated by quantile regression. Excesses of the threshold are modelled with the generalized Pareto distribution (GPD). GAM forms are adopted for the threshold and GPD parameters, and directly estimated—in particular smoothing parameters—by restricted maximum likelihood, which provides an objective and relatively fast method of inference. The GAM models are used to produce return level maps for extreme wind gust speeds over the United States, which show extreme quantiles of the distribution of annual maximum gust speeds. Supplementary materials for this article are available online.

Optimal Threshold Determination for Securities Exchange Volumes Using Improved Maximum Product of Spacing Methodology

To Statisticians, the structure of the extreme levels which exist in the tails of the ordinary distributions is very important in analyzing, predicting and forecasting the likelihood of an occurrence of extreme event. Extreme events are defined as values of the event below or above a certain value called threshold. A well chosen threshold helps to identify the extreme levels. Several methods have been used to determine threshold so as to analyze and model extreme events. One of the most successful methods is the maximum product of spacing (MPS). However, there is a problem encountered while modeling data through this method in that the method breaks down when there is a tie in the exceedances. This study offers a solution to model data even when it contains ties. In the study, a method that improved MPS method for determining an optimal threshold for extreme values in a data set containing ties was derived. The Generalized Pareto Distribution (GPD) parameters for the optimal threshold were derived and compared to GPD parameters determined through the standard MPS model. The study improved the standard MPS methodology by introducing the concept of frequency and used Generalized Pareto Distribution (GPD) and Peak over threshold (POT) methods as the basis of identifying extreme values. The improved MPS models and the standard models were applied to Nairobi Securities Exchange (NSE) trading volume data to determine the GPD parameters for different sectors registered in NSE market and their performance compared. It was realized that the improved MPS model performed better than the standard models. This study will help the Statisticians in different sectors of our economy to model extreme events involving ties.

Optimal Threshold Determination for the Maximum Product of Spacing Methodology with Ties for Extreme Events

Extreme events are defined as values of the event below or above a certain value called threshold. A well chosen threshold helps to identify the extreme levels. Several methods have been used to determine threshold so as to analyze and model extreme events. One of the most successful methods is the maximum product of spacing (MPS). However, there is a problem encountered while modeling data through this method in that the method breaks down when there is a tie in the exceedances. This study offers a solution to model data even if it contains ties. To do so, an optimal threshold that gives more optimal parameters for extreme events, was determined. The study achieved its main objective by deriving a method that improved MPS method for determining an optimal threshold for extreme values in a data set containing ties, estimated the Generalized Pareto Distribution (GPD) parameters for the optimal threshold derived and compared these GPD parameters with GPD parameters determined through the standard MPS model. The study improved maximum product of spacing method and used Generalized Pareto Distribution (GPD) and Peak over threshold (POT) methods as the basis of identifying extreme values. This study will help the statisticians in different sectors of our economy to model extreme events involving ties. To statisticians, the structure of the extreme levels which exist in the tails of the ordinary distributions is very important in analyzing, predicting and forecasting the likelihood of an occurrence of the extreme event.

An automated threshold selection method based on the characteristic of extrapolated significant wave heights

The reasonable estimation of design wave heights is crucial for coastal and offshore engineering practices. To obtain sufficient data for model estimation, the peak over threshold (POT) method with the generalized Pareto distribution (GPD) model is employed. When using this method (i.e., the POT/GPD method), peak excesses over a sufficiently large value (i.e., threshold) are fitted. The extrapolated significant wave heights are highly dependent on the threshold, which must carefully be determined. In this study, significant wave heights from a 40-year (1975–2014) hindcast of tropical cyclone waves in the South China Sea (SCS) are adopted as the initial dataset to study extreme significant wave heights. Based on this initial dataset, the threshold selection is studied by analysing the influence of the excluded samples on the return significant wave heights when the threshold continuously increases. A stable threshold range corresponding to stable return significant wave heights is found, which exhibits a high probability of containing the suitable threshold. To determine the suitable threshold within the stable threshold range, an automated threshold selection method based on the characteristic of extrapolated significant wave heights (ATSME) is proposed. As shown by the reliability (via asymptotic tail approximation and estimation uncertainty) of the POT/GPD method and comparison with other threshold selection methods (i.e., the GPD parameter plot and automated threshold selection method based on the characteristic of the modified scale parameter (ATSMP)), the thresholds obtained by the ATSME are reasonable. Considering that the proposed method is based on the characteristic of extrapolated significant wave heights, data (such as measured, simulated and calculated data) participating in the extrapolation must carefully be processed, which influence the extrapolated results.

Frequency analysis of precipitation extremes under climate change

Frequency analysis of precipitation extremes is significant for the selection of design rainfalls, which are essential inputs for the design of water infrastructure projects, especially when the climate has changed. Therefore, the objective of this study was to propose a framework for more reasonably analysing the frequency of extreme rainfalls. The proposed framework consists of a maximum likelihood estimate (MLE) method for analysing the parameter trends, a hydrological variation diagnosis system to determine abrupt change times, generalized extreme value (GEV) and generalized Pareto distribution (GPD) models for frequency analysis of precipitation extremes, and an ensemble‐methods approach for choosing the most appropriate distributions. The methodology was successfully implemented using a 52‐year time series (1963–2014) of rainfall data recorded by eight rain gauges in Chengbi River basin (south China). The results show that the rainfall series mutated in 1993 and that the entire data set could be divided into two slices (1963–1992 and 1993–2014). Climate change was found to have some impacts on the precipitation extremes: the extreme rainfall value and the parameters of GEV and GPD were variable in the context of climate change. Furthermore, the GPD distribution model outperformed the GEV distribution model.

Using Bayesian Kriging for Spatial Smoothing in Crop Insurance Rating

AbstractRating insurance policies depends on the probability of events in the tail of a distribution. A method to measure such tail‐related risk based on Extreme Value Theory could potentially improve insurance rating. It is also widely agreed that there is a spatial structure to crop yield distributions. Considering the spatial structure may provide more precisely rated policies. In this context, this research provides two contributions in rating area yield crop insurance. One is to provide a method that fits the tail of crop yield distributions using the Generalized Pareto Distribution (GPD), a member of the family of extreme value distributions that models only the tail of the distribution. The second is to estimate parameters of the distribution using a Bayesian Kriging approach that provides spatial smoothing of GPD parameters. The proposed model provides estimates of the spatial structure across regions such as the maximum distance of the spatial effect. Based on an out‐of‐sample performance game between a private insurance company and the federal agency the proposed model provides considerable improvement, particularly when rating deeper tail probability.