From generalized Pareto to extreme values law: Scaling properties and derived features (original) (raw)
Related papers
Communications in Statistics - Theory and Methods, 2018
The Generalized Pareto Distribution (GPD) plays a central role in modelling heavy tail phenomena in many applications. Applying the GPD to actual datasets however is a non-trivial task. One common way suggested in the literature to investigate the tail behaviour is to take logarithm to the original dataset in order to reduce the sample variability. Inspired by this, we propose and study the Exponentiated Generalized Pareto Distribution (exGPD), which is created via log-transform of the GPD variable. After introducing the exGPD we derive various distributional quantities, including the moment generating function, tail risk measures. As an application we also develop a plot as an alternative to the Hill plot to identify the tail index of heavy tailed datasets, based on the moment matching for the exGPD. Various numerical analyses with both simulated and actual datasets show that the proposed plot works well.
International Journal of Statistical Distributions and Applications
Extreme rainfall events have caused significant damage to agriculture, ecology and infrastructure, disruption of human activities, injury and loss of life. They have also significant social, economical and environmental consequences because they considerably damage urban as well as rural areas. Early detection of extreme maximum rainfall helps to implement strategies and measures, before they occur. Extreme value theory has been used widely in modelling extreme rainfall and in various disciplines, such as financial markets, insurance industry, failure cases. Climatic extremes have been analysed by using either generalized extreme value (GEV) or generalized Pareto (GP) distributions which provides evidence of the importance of modelling extreme rainfall from different regions of the world. In this paper, we focus on Peak Over Thresholds approach where the Poisson-generalized Pareto distribution is considered as the proper distribution for the study of the exceedances. This research considers also use of the generalized Pareto (GP) distribution with a Poisson model for arrivals to describe peaks over a threshold. The research used statistical techniques to fit models that used to predict extreme rainfall in Tanzania. The results indicate that the proposed Poisson-GP distribution provide a better fit to maximum monthly rainfall data. Further, the Poisson-GP models are able to estimate various return levels. Research found also a slowly increase in return levels for maximum monthly rainfall for higher return periods and further the intervals are increasingly wider as the return period is increasing.
SN Applied Sciences
The rainfall monitoring allows us to understand the hydrological cycle that not only influences the ecological and environmental dynamics, but also affects the economic and social activities. These sectors are greatly affected when rainfall occurs in amounts greater than the average, called extreme event; moreover, statistical methodologies based on the mean occurrence of these events are inadequate to analyze these extreme events. The Extreme Values Theory provides adequate theoretical models for this type of event; therefore, the Generalized Pareto Distribution (Henceforth GPD) is used to analyze the extreme events that exceed a threshold. The present work has applied both the GPD and its nested version, the Exponential Distribution, in monthly rainfall data from the city of Uruguaiana, in the state of Rio Grande do Sul in Brazil, which calculates the return levels and probabilities for some events of practical interest. To support the results, the goodness of fit criteria is used, and a Monte Carlo simulation procedure is proposed to detect the true probability distribution in each month analyzed. The results show that the GPD and Exponential Distribution fits to the data in all months. Through the simulation study, we perceive that the GPD is more suitable in the months of September and November. However, in January, March, April, and August the, Exponential Distribution is more appropriate, and in the other months, we can use either one.
Applied Mathematical Sciences
Most extreme hydrological events cause severe human and material damage, such as floods and landslides. Extreme rainfall is usually defined as the maximum daily rainfall within each year. In this study, the annual maximum daily rainfalls from 1990 to 2007 are modeled for a station rainfal in Pekanbaru city. The threeparameter generalized extrem value (GEV) and generalizd Pareto (GP) distribution are considered to analized the extrem events. The paramters of these distributions are determined using L-moment method (LMOM). The goodness-offit (GOF) betwen empirical data and theorical distribution are then evaluated. The result shows that GEV provide best fit for station rainfall in Pekanbaru city. Based on the model that have been identified, the return levels of the GEV distribution for station rainfall and their 95% confidence interval are provided. In addition, the return period is also calculated based on the best model in this study, we can reasonably predict the risks associated the extreme event for various return periods. 70 Wenny Susanti et al.
Environment and Ecology Research, 2024
Extreme rainfall events often result in destructive weather conditions, as they frequently lead to flooding. The assessment of return levels, which represent the maximum rainfall that is expected to be exceeded within a given time frame, is crucial for effective flood planning. This study aims to compare the accuracy of return level estimations using two statistical distributions: the stationary Generalized Extreme Value distribution (GEVD) and the stationary Generalized Pareto distribution (GPD). The analysis utilized daily rainfall data from Makassar city, obtained at the Hasanuddin rain gauge station, spanning the period from 1980 to 2022. Two approaches were employed to assess the accuracy of return level estimation: the block maxima (BM) approach with GEVD and the peaks over threshold (POT) approach with GPD. Return levels were estimated for return periods of 2, 3, 4, and 5 years. The root mean square error (RMSE) was used as a metric for comparing the accuracy of the two models. The findings indicate that the GPD outperforms the GEVD in predicting the return level of extreme rainfall for each return period in Makassar city. Furthermore, the study predicts the maximum rainfall expected in the following year. In 2023, based on the GEVD, the maximum rainfall is projected to exceed 144,675 mm/day with a 50% chance of occurrence, while based on the GPD, it is expected to surpass 167,320 mm/day with a 14% chance of occurrence. These predictions provide valuable insights for understanding the potential severity of extreme rainfall events and can assist in planning and managing flood risks in Makassar city.
2016
Extreme value theory (EVT) is a method developed to study extreme events. This method focuses on the behavior of the tail distribution to determine the probability of extreme values. EVT are becoming widely used in various fields of science, such as hydrology, climatology, insurance, and finance. There are two methods to identifying extreme value, Block Maxima (BM) and Peaks Over Threshold (POT). In the case of the univariate approach each methods are follow the Generalized Extreme Value distribution (GEV) and Generalized Pareto Distribution (GPD). Fawcett and Walshaw (2008) defines multivariate extreme as extreme events of a particular variable at several nearby locations (e.g. rainfall over a network of sites). One approach used is based on threshold excess models using bivariate threshold called the Bivariate Generalized Pareto Distribution (BGPD) methods. In this study it will be used BGPD methods with parameter estimation using Maximum Likelihood Estimation, which is then used ...
2010
Previous studies indicate the generalized Pareto distribution (GPD) as a suitable distribution function to reliably describe the exceedances of daily rainfall records above a proper optimum threshold, which should be selected as small as possible to retain the largest sample while assuring an acceptable fitting. Such an optimum threshold may differ from site to site, affecting consequently not only the GPD scale parameter, but also the probability of threshold exceedance. Thus a first objective of this paper is to derive some expressions to parameterize a simple threshold-invariant three-parameter distribution function which is able to describe zero and non zero values of rainfall time series by assuring a perfect overlapping with the GPD fitted on the exceedances of any threshold larger than the optimum one. Since the proposed distribution does not depend on the local thresholds adopted for fitting the GPD, it will only reflect the on-site climatic signature and thus appears particularly suitable for hydrological applications and regional analyses. A second objective is to develop and test the Multiple Threshold Method (MTM) to infer the parameters of interest on the exceedances of a wide range of thresholds using again the concept of parameters threshold-invariance. We show the ability of the MTM in fitting historical daily rainfall time series recorded with different resolutions. Finally, we prove the supremacy of the MTM fit against the standard single threshold fit, often adopted for partial duration series, by evaluating and comparing the performances on Monte Carlo samples drawn by GPDs with different shape and scale parameters and different discretizations.
Symmetry
Several new asymmetric distributions have arisen naturally in the modeling extreme values are uncovered and elucidated. The present paper deals with the extreme value theorem (EVT) under exponential normalization. An estimate of the shape parameter of the asymmetric generalized value distributions that related to this new extension of the EVT is obtained. Moreover, we develop the mathematical modeling of the extreme values by using this new extension of the EVT. We analyze the extreme values by modeling the occurrence of the exceedances over high thresholds. The natural distributions of such exceedances, new four generalized Pareto families of asymmetric distributions under exponential normalization (GPDEs), are described and their properties revealed. There is an evident symmetry between the new obtained GPDEs and those generalized Pareto distributions arisen from EVT under linear and power normalization. Estimates for the extreme value index of the four GPDEs are obtained. In addi...
Journal of Hydrology, 2005
The Generalized Pareto (GP) and Generalized Extreme Value (GEV) distributions have been widely applied in the frequency analysis of numerous meteorological and hydrological events. There are several techniques for the estimation of the parameters, which use the total sample as a source of information. In this paper, we show how valuable estimates are also possible considering only a proper subset of the sample, and we identify the portion of the sample containing the most relevant information for estimating a given parameter. In turn, this may prevent the use of anomalous values, which may adversely affect standard techniques. Here, we illustrate original techniques (based on linear combinations of 'selected' order statistics) to estimate the position parameter, the scale parameter, the quantiles, and the possible scaling behavior of the GP and GEV distributions with negative shape parameters. These estimators are generally unbiased and Mean-Square-Error-consistent. In addition, weakly consistent estimators of quantiles are introduced, the calculation of which does not require the knowledge of any parameter. Some case studies illustrate the applicability of the new techniques in hydrologic practice, and comparisons with standard methods are presented. The new estimators proposed may provide a reasonable alternative to standard methods, and may serve, at least, as a methodology to cross-check the estimates resulting from the application of other techniques.
Trends in Sciences
This paper presents an extension of the generalized extreme value (GEV) distribution, based on the T-X family of distributions: Gompertz-generated family of distributions that make the existing distribution more flexible called the Gompertz-general extreme value (Go-GEV) distribution. Some properties of the proposed distribution are introduced, and a new distribution is applied to actual data, namely rainfall in Lopburi Province, by comparing the proposed model with the traditional GEV distribution and estimating the return levels of the rainfall in Lopburi Province. Results showed that the Go-GEV was an alternative flexible distribution for extreme values that fitted with actual data and described the maximum rainfall better than the traditional GEV distribution. The probability density functions of the Go-GEV distribution had various shapes including left-skewed, right-skewed and close to symmetric. Estimation of the return levels of rainfall values in Lopburi Province by the Go-G...