Testing for a multivariate generalized Pareto distribution

It has recently been shown by Rootzén and Tajvidi (Bernoulli, 12:917–930, 2006) that modelling exceedances of a random variable over a high threshold (peaks-over-threshold approach [POT]) can also in the multivariate setup be done rationally only by a multivariate generalized Pareto distribution (GPD). The selection of a proper threshold is, however, a crucial problem. The contribution of this paper is twofold: We develop first a non asymptotic and exact level-α test based on the single-sample t-test, which checks whether multivariate data are actually generated by a multivariate GPD. Secondly, this procedure is utilized for the derivation of a t-test based threshold selection rule in multivariate peaks-over-threshold models. The application to a hydrological data set illustrates this approach.      

Accounting for threshold uncertainty in extreme value estimation

Tail data are often modelled by fitting a generalized Pareto distribution (GPD) to the exceedances over high thresholds. In practice, a threshold is fixed and a GPD is fitted to the data exceeding . A difficulty in this approach is the selection of the threshold above which the GPD assumption is appropriate. Moreover the estimates of the parameters of the GPD may depend significantly on the choice of the threshold selected. Sensitivity with respect to the threshold choice is normally studied but typically its effects on the properties of estimators are not accounted for. In this paper, to overcome the difficulties of the fixed-threshold approach, we propose to model extreme and non-extreme data with a distribution composed of a piecewise constant density from a low threshold up to an unknown end point and a GPD with threshold for the remaining tail part. Since we estimate the threshold together with the other parameters of the GPD we take naturally into account the threshold uncertainty. We will discuss this model from a Bayesian point of view and the method will be illustrated using simulated data and a real data set.     

An Upper Bound on the Binomial Process Approximation to the Exceedance Process

In this paper we deal with an approximation to the point process of exceedances over a higher threshold. We compute sharp bounds on the remainder terms when actual distributions of exceedances are replaced by appropriate generalized Pareto distributions (GPD). The bound will be formulated in terms of the von Mises function. The ultimate as well as the penultimate approximations are considered; in the latter case the shape parameter of the approximating GPD depends on the threshold.     

A Dynamic Mixture Model for Unsupervised Tail Estimation without Threshold Selection

Exceedances over high thresholds are often modeled by fitting a generalized Pareto distribution (GPD) on R⁺. It is difficult to select the threshold, above which the GPD assumption is enough solid and enough data is available for inference. We suggest a new dynamically weighted mixture model, where one term of the mixture is the GPD, and the other is a light-tailed density distribution. The weight function varies on R⁺ in such a way that for large values the GPD component is predominant and thus takes the role of threshold selection. The full data set is used for inference on the parameters present in the two component distributions and in the weight function. Maximum likelihood provides estimates with approximate standard deviations. Our approach has been successfully applied to simulated data and to the (previously studied) Danish fire loss data set. We compare the new dynamic mixture method to Dupuis' robust thresholding approach in peaks-over-threshold inference. We discuss robustness with respect to the choice of the light-tailed component and the form of the weight function. We present encouraging simulation results that indicate that the new approach can be useful in unsupervised tail estimation, especially in heavy tailed situations and for small percentiles.     

Confidence Intervals and Accuracy Estimation for Heavy-Tailed Generalized Pareto Distributions

The generalized Pareto distribution (GPD) is a two-parameter family of distributions which can be used to model exceedances over a threshold. We compare the empirical coverage of some standard bootstrap and likelihood-based confidence intervals for the parameters and upper p-quantiles of the GPD. Simulation results indicate that none of the bootstrap methods give satisfactory intervals for small sample sizes. By applying a general method of D. N. Lawley, correction factors for likelihood ratio statistics of parameters and quantiles of the GPD have been calculated. Simulations show that for small sample sizes accuracy of confidence intervals can be improved by incorporating the computed correction factors to the likelihood-based confidence intervals. While the modified likelihood method has better empirical coverage probability, the mean length of produced intervals are not longer than corresponding bootstrap confidence intervals. This article also investigates the performance of some bootstrap methods for estimation of accuracy measures of maximum likelihood estimators of parameters and quantiles of the GPD.     

Robust and efficient fitting of the generalized Pareto distribution with actuarial applications in view

Due to advances in extreme value theory, the generalized Pareto distribution (GPD) emerged as a natural family for modeling exceedances over a high threshold. Its importance in applications (e.g., insurance, finance, economics, engineering and numerous other fields) can hardly be overstated and is widely documented. However, despite the sound theoretical basis and wide applicability, fitting of this distribution in practice is not a trivial exercise. Traditional methods such as maximum likelihood and method-of-moments are undefined in some regions of the parameter space. Alternative approaches exist but they lack either robustness (e.g., probability-weighted moments) or efficiency (e.g., method-of-medians), or present significant numerical problems (e.g., minimum-divergence procedures). In this article, we propose a computationally tractable method for fitting the GPD, which is applicable for all parameter values and offers competitive trade-offs between robustness and efficiency. The method is based on ‘trimmed moments’. Large-sample properties of the new estimators are provided, and their small-sample behavior under several scenarios of data contamination is investigated through simulations. We also study the effect of our methodology on actuarial applications. In particular, using the new approach, we fit the GPD to the Danish insurance data and apply the fitted model to a few risk measurement and ratemaking exercises.     

Robust and efficient fitting of the generalized Pareto distribution with actuarial applications in view

Due to advances in extreme value theory, the generalized Pareto distribution (GPD) emerged as a natural family for modeling exceedances over a high threshold. Its importance in applications (e.g., insurance, finance, economics, engineering and numerous other fields) can hardly be overstated and is widely documented. However, despite the sound theoretical basis and wide applicability, fitting of this distribution in practice is not a trivial exercise. Traditional methods such as maximum likelihood and method-of-moments are undefined in some regions of the parameter space. Alternative approaches exist but they lack either robustness (e.g., probability-weighted moments) or efficiency (e.g., method-of-medians), or present significant numerical problems (e.g., minimum-divergence procedures). In this article, we propose a computationally tractable method for fitting the GPD, which is applicable for all parameter values and offers competitive trade-offs between robustness and efficiency. The method is based on ‘trimmed moments’. Large-sample properties of the new estimators are provided, and their small-sample behavior under several scenarios of data contamination is investigated through simulations. We also study the effect of our methodology on actuarial applications. In particular, using the new approach, we fit the GPD to the Danish insurance data and apply the fitted model to a few risk measurement and ratemaking exercises.     

Bayesian approaches for analyzing earthquake catastrophic risk

Extreme value theory has been widely used in analyzing catastrophic risk. The theory mentioned that the generalized Pareto distribution (GPD) could be used to estimate the limiting distribution of the excess value over a certain threshold; thus the tail behaviors are analyzed. However, the central behavior is important because it may affect the estimation of model parameters in GPD, and the evaluation of catastrophic insurance premiums also depends on the central behavior. This paper proposes four mixture models to model earthquake catastrophic loss and proposes Bayesian approaches to estimate the unknown parameters and the threshold in these mixture models. MCMC methods are used to calculate the Bayesian estimates of model parameters, and deviance information criterion values are obtained for model comparison. The earthquake loss of Yunnan province is analyzed to illustrate the proposed methods. Results show that the estimation of the threshold and the shape and scale of GPD are quite different. Value-at-risk and expected shortfall for the proposed mixture models are calculated under different confidence levels.     

广义Pareto模型统计推断及其应用

广义Pareto分布能很好地拟合数据分布的尾部,广泛地应用于金融市场的风险管理、风险经营问题的研究。利用概率加权矩法得到了三参数广义Pareto模型的参数估计式,给出了阈值的选取方法和风险值的计算公式;利用计算机模拟,计算得出了KS检验统计量的临界值。     

GENERALIZED PARETO DISTRIBUTION FIT TO MEDICAL INSURANCE CLAIMS DATA

How to choose an optimal threshold is a key problem in the generalized Pareto distribution （GPD） model. This paper attains the exact threshold by testing for GPD,and shows that GPD model allows the actuary to easily estimate high quantiles and the probable maximum loss from the medical insurance claims data.     

Local polynomial maximum likelihood estimation for Pareto-type distributions

We discuss the estimation of the tail index of a heavy-tailed distribution when covariate information is available. The approach followed here is based on the technique of local polynomial maximum likelihood estimation. The generalized Pareto distribution is fitted locally to exceedances over a high specified threshold. The method provides nonparametric estimates of the parameter functions and their derivatives up to the degree of the chosen polynomial. Consistency and asymptotic normality of the proposed estimators will be proven under suitable regularity conditions. This approach is motivated by the fact that in some applications the threshold should be allowed to change with the covariates due to significant effects on scale and location of the conditional distributions. Using the asymptotic results we are able to derive an expression for the asymptotic mean squared error, which can be used to guide the selection of the bandwidth and the threshold. The applicability of the method will be demonstrated with a few practical examples.     

Approximation of the distribution of excesses using a generalized probability weighted moment method

The POT (Peaks-Over-Threshold) approach consists of using the generalized Pareto distribution (GPD) to approximate the distribution of excesses over a threshold. In this Note, we consider this approximation using a generalized probability weighted moment (GPWM) method. We study the asymptotic behaviour of our new estimators and also the functional bias of the GPD as an estimate of the distribution function of the excesses. To cite this article: J. Diebolt et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

医疗保险索赔数据的广义Pareto分布拟合

How to choose an optimal threshold is a key problemin the generalized Pareto distribution (GPD) model.This paper attains the exactthreshold by testing for GPD,and shows that GPD model allows the actuary to easily estimate high quantiles and the probable maximum loss from the medical insurance claims data.     

Modelling losses and locating the tail with the Pareto Positive Stable distribution

This paper focuses on modelling the severity distribution. We directly model the small, moderate and large losses with the Pareto Positive Stable (PPS) distribution and thus it is not necessary to fix a threshold for the tail behaviour. Estimation with the method of moments is straightforward. Properties, graphical tests and expressions for value-at risk and tail value-at-risk are presented. Furthermore, we show that the PPS distribution can be used to construct a statistical test for the Pareto distribution and to determine the threshold for the Pareto shape if required. An application to loss data is presented. We conclude that the PPS distribution can perform better than commonly used distributions when modelling a single loss distribution for moderate and large losses. This approach avoids the pitfalls of cut-off selection and it is very simple to implement for quantitative risk analysis.     

Modelling losses and locating the tail with the Pareto Positive Stable distribution

This paper focuses on modelling the severity distribution. We directly model the small, moderate and large losses with the Pareto Positive Stable (PPS) distribution and thus it is not necessary to fix a threshold for the tail behaviour. Estimation with the method of moments is straightforward. Properties, graphical tests and expressions for value-at risk and tail value-at-risk are presented. Furthermore, we show that the PPS distribution can be used to construct a statistical test for the Pareto distribution and to determine the threshold for the Pareto shape if required. An application to loss data is presented. We conclude that the PPS distribution can perform better than commonly used distributions when modelling a single loss distribution for moderate and large losses. This approach avoids the pitfalls of cut-off selection and it is very simple to implement for quantitative risk analysis.     

Bayesian Analysis of Extreme Values by Mixture Modeling

Modeling of extreme values in the presence of heterogeneity is still a relatively unexplored area. We consider losses pertaining to several related categories. For each category, we view exceedances over a given threshold as generated by a Poisson process whose intensity is regulated by a specific location, shape and scale parameter. Using a Bayesian approach, we develop a hierarchical mixture prior, with an unknown number of components, for each of the above parameters. Computations are performed using Reversible Jump MCMC. Our model accounts for possible grouping effects and takes advantage of the similarity across categories, both for estimation and prediction purposes. Some guidance on the specification of the prior distribution is provided, together with an assessment of inferential robustness. The method is illustrated throughout using a data set on large claims against a well-known insurance company over a 15-year period.     

A Predictive Approach to Tail Probability Estimation

One of the issues contributing to the success of any extreme value modeling is the choice of the number of upper order statistics used for inference, or equivalently, the selection of an appropriate threshold. In this paper we propose a Bayesian predictive approach to the peaks over threshold method with the purpose of estimating extreme quantiles beyond the range of the data. In the peaks over threshold (POT) method, we assume that the threshold identifies a model with a specified prior probability, from a set of possible models. For each model, the predictive distribution of a future excess over the corresponding threshold is computed, as well as a conditional estimate for the corresponding tail probability. The unconditional tail probability for a given future extreme observation from the unknown distribution is then obtained as an average of the conditional tail estimates with weights given by the posterior probability of each model.     

Clusters of extremes: modeling and examples

We study clusters of threshold exceedances caused by dependence in time series. The clusters are defined as conglomerates containing consecutive threshold exceedances of the series separated by return intervals with consecutive non-exceedances. We derive asymptotic distributions of the cluster and inter-cluster sizes for processes with the extremal index equal to zero, the asymptotic expectation of the inter-cluster size and an exponential rate of convergence of the distribution tail of the return interval between clusters to the stable distribution tail. Distributions of the cluster and inter-cluster sizes of ARMAX, MM and AR(1) processes are obtained.     

Estimation of Parameters of the Generalized Pareto\Distribution by the Least Squares

Traditional estimations of parameters of the generalized Pareto distribution (GPD) are generally constrained by the shape parameter of GPD. Such as: the method-of-moments (MOM), the probability-weighted moments (PWM), L-moments (LM), the maximum likelihood estimation (MLE) and so on. In this paper we use the fact that GPD can be transformed into the exponential distribution and use the results of parameters estimation for the exponential distribution, than we propose parameters estimators of the two-parameter or three-parameter GPD by the least squares method. Some asymptotic results are provided and the proposed method not constrained by the shape parameter of GPD. A simulation study is carried out to evaluate the performance of the proposed method and to compare them with other methods suggested in this paper. The simulation results indicate that the proposed method performs better than others in some common situation.     

广义Pareto分布的广义有偏概率加权矩估计方法

广义Pareto分布(GPD)是统计分析中一个极为重要的分布,被广泛应用于金融、保险、水文及气象等领域.传统的参数估计方法如极大似然估计、矩估计及概率加权矩估计方法等已被广泛应用,但使用中存在一定的局限性.虽然提出很多改进方法如广义概率加权矩估计、L矩和LH矩法等,但都是研究完全样本的估计问题,而在水文及气象等应用领域常出现截尾样本.本文基于概率加权矩理论,利用截尾样本对三参数GPD提出一种应用范围广且简单易行的参数估计方法,可有效减弱异常值的影响.首先求解出具有较高精度的形状参数的参数估计,其次得出位置参数及尺度参数的参数估计.通过Monte Carlo模拟说明该方法估计精度较高.