A new hybrid estimation method for the generalized pareto distribution

The study of extremes has attracted the attention of scientists, engineers, actuaries, policy makers, and statisticians for many years. Extreme value theory (EVT) deals with the extreme deviations from the median of probability distributions and is used to study rare but extreme events. EVT’s main results characterize the distribution of the sample maximum or the distribution of values above a given threshold. In this study, EVT has been used to construct a model on the extreme and rare earthquakes that have happened in the United States from 1700 to 2011.The primary goal of fitting such a model is to estimate the amount of losses due to those extreme events and the probabilities of such events. Several diagnostic methods (for example, QQ plot and Mean Excess Plot) have been used to justify that the data set follows generalized Pareto distribution (GPD). Three estimation techniques have been employed to estimate parameters. The consistency and reliability of estimated parameters have been observed for different threshold values. The purpose of this study is manifold: first, we investigate whether the data set follows GPD, by using graphical interpretation and hypothesis testing. Second, we estimate GPD parameters using three different estimation techniques. Third, we compare consistency and reliability of estimated parameters for different threshold values. Last, we investigate the bias of estimated parameters using a simulation study. The result is particularly useful because it can be used in many applications (for example, disaster management, engineering design, insurance industry, hydrology, ocean engineering, and traffic management) with a minimal set of assumptions about the true underlying distribution of a data set. KEYWORDS: Extreme Value Theory; QQ Plot; Mean Excess Plot; Mean Residual Plot; Peak Over Threshold; Generalized Pareto Distribution; Maximum Likelihood Method; Method of Moments; Probability-Weighted Moments; Shapiro-Wilk test; Anderson- Darling Test

As part of the flood risk assessment in The Netherlands, extreme hydraulic boundary conditions (HBC) at coastal water defenses are calculated to assess the safety and sufficiency of the defenses.For this purpose, extreme offshore values of wave height and wave period with a frequency of occurrence of 1/10,000 years need to be estimated.The estimation is carried out using measurements at nine offshore locations in the North Sea, with a record length of 30 years.To estimate extreme wave heights and wave periods, the Generalized Pareto Distribution (GPD), which follows from extreme value theory, was fitted to peaks-over-threshold (POT) at each of the nine measurement stations.The fitted GPD parameters and consequently the extrapolated extreme values are particularly sensitive to the choice of threshold.A reproducible method was employed for threshold selection based on the anticipated behavior of the GPD shape parameter as the threshold approaches the domain of attraction.A modified regional frequency analysis (MRFA) was subsequently carried out to spatially smooth the GPD shape parameters.Final GPD fits were made using the shape-parameter output of the MRFA to produce final extrapolated extreme value estimates.The extreme values were compared with those resulting from an operationally-used distribution function in The Netherlands known as the conditional Weibull distribution.

Aim of study: The aim of this work was to model diameter distributions of Quercus suber stands. The ultimate goal was to construct models enabling the development of more affordable forest inventory methods. This is the first study of this type on cork oak forests in the area.Area of study: The area of study is “Los Alcornocales” Natural Park (Cádiz-Málaga, Spain).Material and methods: The diameter distributions of 100 permanent plots were modelled with the two-parameter Weibull function. Distribution parameters were fitted with the non-linear regression, maximum likelihood, moment and percentile-based methods. Goodness of fit with the different methods was compared in terms of number of plots rejected by the Kolmogorov-Smirnov test, bias, mean square error and mean absolute error. The scale and shape parameters in the Weibull function were related to the stand variables by using the parameter prediction model.Main results: The best fitting was obtained with the non-linear regression approach, using as initial values those obtained by maximum likelihood method, the percentage of rejections by the Kolmogorov-Smirnov test was 2% of the total number of cases. The scale parameter (b) was successfully modelled in terms of the quadratic mean diameter under cork (R2 adj = 0.99). The shape parameter (c) was modelled by using maximum diameter, minimum diameter and plot elevation (R2 adj = 0.40).Research highlights: The proposed model diameter distribution can be a highly useful tool for the inventorying and management of cork oak forests.Key words: maximum likelihood method; moment method; non linear regression approach; parameter prediction model; percentile method; scale parameter; shape parameter.

Abstract. The evaluation of the probability of occurrence of extreme natural events is important for the protection of urban areas, industrial facilities and others. Traditionally, the extreme value theory (EVT) offers a valid theoretical framework on this topic. In an over-threshold modelling (OTM) approach, Pickands' theorem, (Pickands, 1975) states that, for a sample composed by independent and identically distributed (i.i.d.) values, the distribution of the data exceeding a given threshold converges through a generalized Pareto distribution (GPD). Following this theoretical result, the analysis of realizations of environmental variables exceeding a threshold spread widely in the literature. However, applying this theorem to an auto-correlated time series logically involves two successive and complementary steps: the first one is required to build a sample of i.i.d. values from the available information, as required by the EVT; the second to set the threshold for the optimal convergence toward the GPD. In the past, the same threshold was often employed both for sampling observations and for meeting the hypothesis of extreme value convergence. This confusion can lead to an erroneous understanding of methodologies and tools available in the literature. This paper aims at clarifying the conceptual framework involved in threshold selection, reviewing the available methods for the application of both steps and illustrating it with a double threshold approach.

The two parametric distribution functions appearing in the extreme-value theory – the generalized extreme-value distribution and the generalized Pareto distribution – have log-concave densities if the extreme-value index γ∈[−1, 0]. Replacing the order statistics in tail-index estimators by their corresponding quantiles from the distribution function that is based on the estimated log-concave density ˆ f n leads to novel smooth quantile and tail-index estimators. These new estimators aim at estimating the tail index especially in small samples. Acting as a smoother of the empirical distribution function, the log-concave distribution function estimator reduces estimation variability to a much greater extent than it introduces bias. As a consequence, Monte Carlo simulations demonstrate that the smoothed version of the estimators are well superior to their non-smoothed counterparts, in terms of mean-squared error.

Current estimates of the European windstorm climate and their associated losses are often hampered by either relatively short, coarse resolution or inhomogeneous datasets. This study tries to overcome some of these shortcomings by estimating the European windstorm climate using dynamical seasonal-to-decadal (s2d) climate forecasts from the European Centre for Medium-Range Weather Forecasts (ECMWF). The current s2d models have limited predictive skill of European storminess, making the ensemble forecasts ergodic samples on which to build pseudoclimates of 310–396 yr in length. Extended winter (October–April) windstorm climatologies are created using scalar extreme wind indices considering only data above a high threshold. The method identifies up to 2363 windstorms in s2d data and up to 380 windstorms in the 40-yr ECMWF Re-Analysis (ERA-40). Classical extreme value analysis (EVA) techniques are used to determine the windstorm climatologies. Differences between the ERA-40 and s2d windstorm climatologies require the application of calibration techniques to result in meaningful comparisons. Using a combined dynamical–statistical sampling technique, the largest influence on ERA-40 return period (RP) uncertainties is the sampling variability associated with only 45 seasons of storms. However, both maximum likelihood (ML) and L-moments (LM) methods of fitting a generalized Pareto distribution result in biased parameters and biased RP at sample sizes typically obtained from 45 seasons of reanalysis data. The authors correct the bias in the ML and LM methods and find that the ML-based ERA-40 climatology overestimates the RP of windstorms with RPs between 10 and 300 yr and underestimates the RP of windstorms with RPs greater than 300 yr. A 50-yr event in ERA-40 is approximately a 40-yr event after bias correction. Biases in the LM method result in higher RPs after bias correction although they are small when compared with those of the ML method. The climatologies are linked to the Swiss Reinsurance Company (Swiss Re) European windstorm loss model. New estimates of the risk of loss are compared with those from historical and stochastically generated windstorm fields used by Swiss Re. The resulting loss-frequency relationship matches well with the two independently modeled estimates and clearly demonstrates the added value by using alternative data and methods, as proposed in this study, to estimate the RP of high RP losses.

Extreme value analysis for waves in the Persian Gulf: Skill assessment of different methods for a fetch-limited basin

PurposeThis study aims to examine the stock returns distributions in ten countries in the periods before and after the global financial crisis (GFC) to evaluate how well the empirical distributions conformed to the extreme value theory (EVT) which underlies a family of risk management models.Design/methodology/approachThe authors’ sample consists of the G5 countries and the five leading emerging economies. Parameters of the General Pareto Distribution (GPD) for each country are estimated for the pre‐ and the crisis period. The authors follow a two‐step procedure: a GARCH(1,1) model is fitted to the historical return data by pseudo maximum likelihood method; Hill's GPD tail estimation procedure is employed on the residuals from the first step. Goodness‐of‐fit is evaluated for the empirical distributions.FindingsThe authors find that the EVT explains the observed distributions well in both the pre‐GFC and the GFC periods, with the important exceptions of the US and the UK markets in the crisis period. Moreover, the estimated distribution parameters are quite different for the two periods. The results underscore the inadequacy of the quantitative risk models in times of financial turbulence, and the need for prudential exercise of judgment in risk management.Originality/valueThe global financial crisis (GFC) provides a unique and historical experiment to evaluate the models of tail distributions. Although the EVT provides a sound basis for modeling extreme risks, the study highlights the fundamental problem of dealing with uncertainty.

Climate change has brought about unprecedented new weather patterns, one of which is changes in extreme rainfall. In Kenya, heavy rains and severe flash floods have left people dead and displaced hundreds from their settlements. In order to build a resilient society and achieve sustainable development, it is paramount that adequate inference about extreme rainfall be made. To this end, this research modelled and predicted extreme rainfall events in Kenya using Extreme Value Theory for rainfall data from 1901-2016. Maximum Likelihood Estimation was used to estimate the model parameters and block maxima approach was used to fit the Generalized Extreme Value Distribution (GEVD) while the Peak Over Threshold method was used to fit the Generalized Pareto Distribution (GPD). The Gumbel distribution was found to be the optimal model from the GEVD while the Exponential distribution gave the optimal model over the threshold value. Furthermore, prediction for the return periods of 10, 20, 50 and 100 years were made using the return level estimates and their corresponding confidence intervals were presented. It was found that increase in return periods leads to a corresponding increase in return levels. However, the GPD gave higher return levels for 10 and 20 years compared to GEVD. While, for higher return periods 50 and 100 years, the GEVD gave higher return levels compared to the GPD. Model diagnostics using probability, density, quantile and return level plots indicated that the models provided were a good fit for the data.

High volatility, thick tails and extreme value theory in value-at-risk estimation

This paper aims to model the occurrence of daily precipitation extreme events and to estimate the return period of these events through the extreme value theory (generalized extreme value distribution (GEV) and the generalized Pareto distribution (GPD)). The GEV and GPD were applied in precipitation series of homogeneous regions of the Brazilian Amazon. The GEV and GPD goodness of fit were evaluated by quantile–quantile (Q-Q) plot and by the application of the Kolmogorov–Smirnov (KS) test, which compares the cumulated empirical distributions with the theoretical ones. The Q-Q plot suggests that the probability distributions of the studied series are appropriated, and these results were confirmed by the KS test, which demonstrates that the tested distributions have a good fit in all sub-regions of Amazon, thus adequate to study the daily precipitation extreme event. For all return levels studied, more intense precipitation extremes is expected to occur within the South sub-regions and the coastal area of the Brazilian Amazon. The results possibly will have some practical application in local extreme weather forecast.

Felsenstein, J. (Department of Genetics SK-50, University of Washington, Seattle, Washington 98195). 1973. Maximum likelihood and minimum-steps methods for estimating evolutionary trees from data on discrete characters. Syst. Zool. 22:240-249.The general maximum likelihood approach to the statistical estimation of phylogenies is outlined, for data in which there are a number of discrete states for each character. The details of the maximum likelihood method will depend on the details of the probabilistic model of evolution assumed. There are a very large number of possible models of evolution. For a few of the simpler models, the calculation of the likelihood of an evolutionary tree is outlined. For these models, the maximum likelihood tree will be the same as the parsimonious (or minimum-steps) tree if the probability of change during the evolution of the group is assumed a priori to be very small. However, most sets of data require too many assumed state changes per character to be compatible with this assumption. Farris (1973) has argued that maximum likelihood and parsimony methods are identical under a much less restrictive set of assumptions. It is argued that the present methods are preferable to his, and a counterexample to his argument is presented. An algorithm which enables rapid calculation of the likelihood of a phylogeny is described. [Evolutionary trees: maximum likelihood.] The first systematic attempt to apply standard statistical inference procedures to the estimation of evolutionary trees was the work of Edwards and Cavalli-Sforza (1964; see also Cavalli-Sforza and Edwards, 1967). At about the same time, the parsimony' or minimum evolutionary steps method of Camin and Sokal (1965) gave a great impetus to the development of welldefined procedures for obtaining evolutionary trees. Edwards and Cavalli-Sforza concerned themselves with data from continuous variables such as gene frequencies and quantitative characters. The CaminSokal approach, on the other hand, was developed for characters which are recorded as a series of discrete states. Although some taxonomists have declared that the problem of guessing phylogenies should be viewed as a problem of statistical inference (Farris, 1967, 1968; Throckmorton, 1968), until recently there have been no attempts to explore the relationship between the statistical inference and minimum-steps approaches. Recently, Farris (1973) has presented a detailed argument that, under certain reasonable assumptions, the maximum-likelihood method of statistical inference appropriate to discrete-character data is precisely the parsimony method of Camin and Sokal. In this paper, I will examine the application of maximum likelihood methods to discrete characters, and will show that parsimony methods are not maximum likelihood methods under the assumptions made by Farris. They are maximum likelihood methods under considerably more restrictive assumptions about evolution. METHODS OF MAXIMUM LIKELIHOOD Suppose that we want to estimate the evolutionary tree, T, which is to be specified by the topological form of the tree and the times of branching. We are given a set of data, D, and a model of evolution, M, which incorporates not only the evolutionary processes, but also the processes of sampling by which we obtained the data. This model will usually be probabilistic, involving random events such as changes of the environment, occurrence of favorable

The statistics of reverberation in active sonar are characterized by non-Rayleigh amplitudes in the normalized matched filter output. Unaccounted for, this property can lead to high false alarm rates in fixed-threshold detectors. A new approach to modeling threshold-crossing statistics based on extreme value theory is proposed which uses the generalized Pareto distribution as the unique, asymptotic model of the tail distribution, valid at large thresholds. Methods of parameter estimation are discussed and applied to active sonar reverberation collected on a hull-mounted sonar system. The statistics of reverberation in active sonar are found to generally have a power-law behavior in the tails with a shape parameter that is persistent in time and bandwidth dependent. The threshold needed for accurate parameter estimation is generally found to be well below that of typical fixed-threshold detectors.

This case study focuses on modeling the real, unique data set of 4245 operational risk claims of an anonymous Central and Eastern European insurance company from 2010 to 2018. We apply extreme value theory to build a more complex model, estimating losses from operational risk events using available historical claims. Low frequency, high-severity claims are identified using the extreme value theory peak-over-threshold method, and modeled via the generalized Pareto distribution, and compared with with Frechet, Weibull and Gumbel distributions. The compound lognormal distribution is used for high-frequency, low-severity claims. Using the bootstrapping principle, many one-year claims portfolio predictions are generated to calculate value-at-risk. The aim of this paper is to examine the sufficiency of the standard formula approach as defined in Solvency II in the case of a company with a specific risk profile. Based on the calculations for the real, unique data set we develop a new approach for estimating the solvency capital requirement for operational risk. Our findings show that the application of extreme value theory to internal data provides a more conservative capital requirement estimation than the standard formula calculation. Thus, the analyzed insurance company may be vulnerable to potential losses when the standard formula is applied.

Impact of thresholds on nonstationary frequency analyses of peak over threshold extreme rainfall series in Pearl River Basin, China