Generalized Pareto Distribution Research Articles

CHASING THE TAIL: A GENERALIZED PARETO DISTRIBUTION APPROACH TO ESTIMATING WEALTH INEQUALITY

AbstractSince the work reported in Vermeulen (Review of Income and Wealth, 64(2), 357–387; 2018), a literature has developed using the simple Pareto distribution along with “rich list” information to improve estimates of the upper tail of the wealth distribution measured in surveys. Because the construction of such external data is typically opaque and subject to error, it may be best not to depend exclusively on this approach. This paper develops an alternative approach, using the generalized Pareto distribution (GPD), extending an estimation strategy developed by Castillo and Hadi (Journal of the American Statistical Association, 92(440), 1609–1620; 1997) in a novel way to survey data. The extension allows for addressing imperfections in the effective coverage of the upper tail of the wealth distribution, as well as allowing the imposition of an external measure of aggregate wealth as a constraint. The paper implements the method on Austrian survey data that has weak coverage of wealthy households. The results indicate that even with a range of plausible adjustments for data imperfections, relying only on the survey data to project the upper tail is insufficient to yield implied wealth aggregates in the range implied by aggregate figures. Imposing the implied aggregate as a constraint is necessary. The approach may be particularly useful in comparing survey‐based inequality measures across and within countries and over time, as well as in constructing distributional financial accounts.

A Credibility Framework for Extreme Value-at-Risk

Value-at-risk estimates derived from extreme value data by fitting fat-tailed distributions can be so large that their validity is open to question. In this paper, an objective criterion, and a framework from which it was developed, are presented in order to decide whether or not a fitted distribution is inappropriate for the purpose of value-at-risk calculation. That criterion is based on established extreme value theory (principally the Pickands-Balkema-deHaan Theorem), which is used to calculate a sequence of reference value-at-risk estimates using Generalised Pareto distributions. Those estimates are used to develop a closed-form formula for calculating a theoretical ’maximum’ value-at-risk. The method is validated by generating 100 random data sets and testing them against the framework for varying input parameter values. Approximately 75% of those cases passed the validation test.

The Lindley generalized Pareto distribution: properties, estimation and application to COVID-19 data

The Lindley generalized Pareto distribution: properties, estimation and application to COVID-19 data

Study on the distribution of average wind speeds at a mountainous bridge site for structural durability design

Conventional wind speed distribution methods (e.g., Rayleigh distribution and Weibull distribution) may not adequately capture complex characteristics of wind fields in mountainous areas. To address this problem, this study proposes a semi-parametric mix method for modeling the distribution of average wind speeds based on the combination of nonparametric Kernel Density Estimation (KDE) and Generalized Pareto Distribution (GPD). In the proposed method, KDE focuses on capturing the distribution in the main part of average wind speeds, while GPD aims at performing the distribution in terms of those in the extreme part. The segment point (i.e., the threshold) between KDE and GPD distributions is determined based on the combination of conditional mean excesses criterion and empirical rule. Meanwhile, the selection of modeling parameters should ensure that the mix distribution model is continuous and differentiable at the identified threshold point. Then, the commonly-used conditional probability model is further introduced to describe the wind direction distribution. Finally, a case study based on the measured 10-min average wind speeds at a mountainous bridge site is employed to demonstrate the effectiveness of the proposed method. The results indicate that: (1) the distribution of omnidirectional average wind speeds in the mountainous bridge site exhibits an obviously single-peak characteristic, while those considering wind directionality present a certain bimodal characteristic; (2) the proposed method can effectively describe wind speed distributions with different statistical characteristics, and the fitting accuracy outperforms the frequently-employed Weibull distribution model.

Impact of Drought in Northeastern Algeria: Comparative Study of the SPI and SPEI Índices

Purpose: The study aimed to evaluate drought trends in Annaba, Algeria, considering global warming and extreme weather, using quantitative indices for improved drought comprehension and prediction. Theoretical Framework:The study assesses changes in drought frequency and severity in the Annaba region of Algeria due to global warming and extreme weather conditions. It employs quantitative indices to enhance drought understanding and prediction. Methodology: The study analyzes temperature and precipitation data (1981-2021) from an Annaba weather station to assess drought characteristics.Standardized indices (SPI, SPEI) and extreme value analysis (Generalized Pareto Distribution) are used to quantify drought intensity, duration, and temperature impacts. Findings: Our analyses reveal a critical divergence in drought assessment tools. While stable monthly temperatures suggest a constant influence on evapotranspiration, the SPEI index (including temperature) indicates increasing drought compared to the precipitation-only SPI. This highlights the importance of considering temperature alongside precipitation for accurate drought assessment. Research Practical and Social Implication: This study emphasizes the significance of considering the interplay among temperature, precipitation, and evapotranspiration for better understanding and predicting drought changes in the Annaba region, Algeria. These insights are crucial for sustainable water resource management and climate change adaptation in the region. Originality/Value:This study distinguishes itself by its comprehensive analysis of long-term climate data to examine drought trends in a specific region (Annaba, Algeria) severely impacted by climate change. By comparing the effectiveness of two drought indices (SPEI and SPI), it offers valuable insights into how to improve drought assessments by considering the role of temperature in evapotranspiration. This research enriches the existing literature on drought management and climate change adaptation, providing a valuable case study for similar arid and semi-arid regions worldwide.

Graphical Diagnostics for Threshold Selection in Fitting the Generalized Pareto Distribution

Graphical Diagnostics for Threshold Selection in Fitting the Generalized Pareto Distribution

Addressing the financial impact of natural disasters in the era of climate change

The objective of our study is to predict the financial losses that may result from natural disasters, along with their level of volatility, over a period of 1 to 15 years. Volatility can lead to significant fluctuations in Profit and Loss (P&L) for companies that are affected by unexpected events. To achieve this goal, we created a novel two-factor square-root model that allows us to establish a correlation between the frequency of occurrences and volatility, using correlated Brownian motions. Moreover, we utilized a Generalized Pareto Distribution (GPD) to estimate the maximum potential loss in terms of Value at Risk (VaR) for each specific type of natural disaster. To ensure the reliability of our predictions, we compared our results to those of four reference models and conducted a backtesting analysis. This approach is particularly suitable for insurance companies seeking to maintain stable reserves, but it can also be adapted for any other type of business that is vulnerable to extreme events and aims to safeguard a consistent cash flow for their stakeholders.

Transformer encoder based self-supervised learning for HVAC fault detection with unlabeled data

Data driven methods are the most studied fault detection and diagnostics (FDD) type in buildings HVAC systems. However, most studies rely on labeled data for specific faults which are hard to find and collect for real systems. While the fault-free data is easier to collect, it is still time consuming to label for large systems operation. Moreover, most of the studies rely on the usage of supervised learning algorithms which do not generalize well beyond the training data making unseen faults hard to detect. In this paper, we define a methodology to use a self-supervised learning method for HVAC systems' FDD using a Transformer encoder, moreover, we tested it on a real case study. By strategically masking portions of the multivariate time-series data using Markov chain approach with two states. The model is trained by predicting these concealed segments. This approach, independent of labeled data, offers a scalable solution for practical HVAC applications. Anomalies are labeled using the Peak Over Threshold (POT) method, which dynamically determines thresholds by fitting reconstruction errors to a generalized Pareto distribution. Subsequent fault diagnostics emphasize features with pronounced reconstruction errors, pinpointing potential HVAC malfunctions. This methodology reduces dependence on labeled datasets and augments the model's generalization, facilitating detection of unobserved faults. This approach was applied to data from a real building. As a results multiple faults were detected mainly due to the malfunctioning of the monitoring system. The model demonstrates the ability to detect both sequential and individual faults. The period from October 19th to December 23rd was detected as a fault period due to the change in the trend of the data because of the monitoring system.

A study on wave climate variability along the nearshore regions of Bohai Sea based on long term observation data

The Bohai Sea off the coast of northern China plays important economic and ecological roles. No studies have been conducted on wave characteristics in the Bohai Sea nearshore area using ocean station data in the past. Observational data (1998–2022) from eight ocean-stations in the Bohai Sea were used in the present study. Overall, wave height (Hs) was small; winter seasons had higher Hs than summer seasons, and the maximum Hs occurred in the Bohai Strait. Joint distribution analysis indicated that waves with Hs ranging from 0.2 to 0.4 m, and mean wave periods (Tm) ranging from 3 to 3.5 s, were most common. Hs in the Bohai Sea nearshore area showed a weak response to climate variability and was moderately correlated with the Ocean Niño Index (ONI) and the North Pacific Index (NPI) in winter, and with the Arctic Oscillation (AO) in summer. Under extreme conditions, Hs was best described by a Generalized Pareto distribution, Tm was best described by a GEV distribution, and wave directions mainly followed a t location-scale distribution. The maximum wave height (Hmax)/Hs was 1.51, and the ratio varied with location. Machine learning methods were used to calculate Hmax more precisely, and they yielded better results than statistical methods.

Connecting flow duration curve and precipitation duration curve based on the relationship deduced from machine learning in the watersheds of northern China

Obtaining reliable hydrological data for ungauged watersheds has always been a significant challenge in the water resources management. The flow duration curve (FDC) and precipitation duration curve (PDC) are classical methods to address this challenge. However, traditional FDC and PDC methods mostly rely on pre-assumed probability distribution function form, which may result in sub-optimal model parameters and consequently sub-optimal simulation results of hydrological data, and the common function forms are not applicable to deal with the data including zero-values, which often occurs in the discharge data of arid region and daily rainfall data. To address the issues in FDC and PDC , this study firstly determines the most suitable function form to describe FDC and PDC from various distribution functions, i.e., Log Normal, Generalized Pareto Distribution and H2018 function, from 1960 to 2010 in 7 watersheds in the north of China. The multiple methods including Random Forest, polynomial, Multilayer Perceptron and K-Nearest Neighbor are employed to simulate the parameters of the functions, aiming to deduce the optimal parameters of FDC functions. The results show that, H2018 function form has the best performance in simulating the PDC and FDC and in general, parameter a and b of H2018 function indicate an opposite trend, and parameter k declines in mostly condition at the annual scale from 1960 to 2010. The results of different methods indicate that the Random Forest method is the most efficient and accurate approach to deduce FDC parameters from measured precipitation and climate factors, compared with polynomial, Multilayer Perceptron and K-Nearest Neighbor. Based on the relationship between PDC parameters and FDC parameters, the FDC of ungauged basins can be deduced from the observed precipitation data. The research findings provide a new method to estimate FDC for the water resources management in the ungauged basins.

Return Levels on Stationary Extreme Rainfall Series: A Comparative Study of Generalized Extreme Value and Generalized Pareto Distributions

Return Levels on Stationary Extreme Rainfall Series: A Comparative Study of Generalized Extreme Value and Generalized Pareto Distributions

The impact of the tail risk of demand on corporate investment: Evidence from Chinese manufacturing firms

The tail risk of demand over the past certain periods not only affect entrepreneurs' expectations on future demand uncertainty, increase the option value of “waiting an see” for delay investment, but also affect entrepreneurs' confidence and risk appetite by adjusting the worst-case scenario in the “demand belief set”, thereby further inhibiting investment. Utilizing the year-on-year sales growth data of various manufacturing industries in China, and the General Pareto Distribution method of Extreme Value Theory to estimate tail-index parameter and tail threshold as proxies for the tail risk of industry-level demand. Combined with the firm level data, our empirical results show that the tail risk indeed affect firm investment negatively, especially after the outbreak of global financial crisis, but such focal effect is stronger for firms with non-state ownership, small size, lower productivity and stricter financing constraints. Expansionary macroeconomic policies such as monetary and fiscal policy in periods of high tail risk have a higher incentive effect on firm investment, which provides more evidence for the effectiveness of macroeconomic policies during the economic recessions.

Modeling rainfall extremes along the coastal and Northern parts of Ghana

The main objective of the study was to determine the appropriate distribution for extreme rainfall along the coastal and northern sectors of Ghana. For stakeholders and policymakers to make appropriate risk-mitigating measures to lessen the damage caused by flood and drought, it is necessary to make proper inferences about extreme rainfall. In this study, we used both the multivariate and univariate extreme value data analysis approaches. The Generalized Extreme Value (GEV) with the Block Maxima approach and Generalized Pareto Distribution (GPD) with the Peak over the threshold (that is all excesses and decluster peaks approaches) were used in this study. Historical gridded monthly maximum rainfall data from 1970 to 2020 were obtained from the Climatic Research Unit and were grouped as the coastal and northern stations. The Maximum Likelihood Estimation method was used to estimate the model parameters, and both the unit root test and the Mann-Kendall tests were used to test for trend in the data. With the multivariate extreme modelling approach, the logistic bivariate GEV model was chosen as the “best” model. However, the dependence value was 0.965, so the extreme rainfall should be modelled independently using the univariate extreme value approaches. Hence, based on the information criteria and analysis of deviance approaches, the GEV distribution was considered the “best” fit for the extreme rainfall dataset for the northern part of Ghana. In contrast, the GPD distribution was the “best” fit for the coastal station. Comparatively, for the volume of rainfall in the year 2020, the extreme rainfall is expected to be higher in the coastal station of Ghana in the next two years. Also, extreme rainfall in 2 years would not exceed the maximum occurrence of rainfall (279.267), which happened in September 2020 at the northern station of Ghana.

Impacts and Risk Assessments of Climate Change for the Yields of the Major Grain Crops in China, Japan, and Korea.

Climate change poses a high risk to grain yields. Maize, rice, and wheat are the three major grain crops in China, Japan, and Korea. Assessing the impacts and risks of climate on the yields of these grain cops is crucial. An economy-climate model (C-D-C model) was established to assess the impacts of climate factors on the grain yields in different crop areas. The peaks over threshold model based on the generalized Pareto distribution was used to calculate the value at risk and the expected shortfall, which can evaluate the yield risk of different crops. The impact ratio of climate change was employed to estimate the impacts of climate change under different climate scenarios. The main conclusions can be summarized as follows: the impacts of climate factors on grain yields and the risk vary widely across the different regions and crops. Compared to 1991-2020, climate change from 2021 to 2050 exerts positive impacts on rice and wheat, while the negative impacts on maize in the crop areas are significantly affected by climate factors. The impact ratios of climate change are larger in the SSP1-2.6 and the SSP5-8.5 scenarios than under the SSP2-4.5 scenario. These findings are useful for targeting grain yields in smaller study areas.

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.

Inference for New Environmental Contours Using Extreme Value Analysis

AbstractEnvironmental contours are often used in engineering applications to describe risky combinations of variables according to some definition of an exceedance probability. These contours can be used to both understand multivariate extreme events in environmental processes and mitigate against their effects, e.g. in the design of structures. Such ideas are also useful in other disciplines, with the types of extreme events of interest depending on the context. Despite clear connections with extreme value modelling, much of this methodology has so far not been exploited in the estimation of environmental contours; in this work, we provide a way to unify these areas. We focus on the bivariate case, introducing two new definitions of environmental contours. We develop techniques for their inference which exploit a non-standard radial and angular decomposition of the variables, building on previous work for estimating limit sets. Specifically, we model the upper tails of the radial distribution using a generalised Pareto distribution, with adaptable smoothing of the parameters of this distribution. Our methods work equally well for asymptotically independent and asymptotically dependent variables, so do not require us to distinguish between different joint tail forms. Simulations demonstrate reasonable success of the estimation procedure, and we apply our approach to an air pollution data set, which is of interest in the context of environmental impacts on health.Supplementary materials accompanying this paper appear online.

Nonstationary coastal flood hazard analysis

AbstractIt is well known that the increase in coastal flood hazard is expected to continue due primarily to climate change. While, previous studies have advanced probabilistic approaches for estimating future coastal flood hazard, two seminal issues have received little attention: (1) sea level trend projections that arise as a consequence of climate change not being incorporated in nonstationary coastal flood frequency analysis and (2) traditional stationary approaches that estimate return periods are not suited for analysis of nonstationary water levels. In this paper, a regression approach for sea level rise estimation is proposed to detrend water levels. The peak-over-threshold approach is used to analyze the probabilistic behavior of detrended water levels via a generalized Pareto distribution. Two interpretations of return period—the expected waiting time until an exceedance event occurs and the time associated with the expected number of exceedance events equal to one—are elaborated in a nonstationary context. By incorporating the existing sea level trend projections and using two interpretations of return period, flood hazard curves with uncertainty (i.e., water levels vs. return periods) are developed to describe the probabilistic behavior of future coastal flood hazards. Two case studies are conducted: one for Boston, Massachusetts, and one for New York City, New York.

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.

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.