Estimation of the angular density in bivariate generalized Pareto models

We investigate a method to estimate the angular density non-parametrically in bivariate generalized Pareto models. The angular density can be used as a visual tool to gain a first insight into the tail-dependence structure of given data. We derive a representation of the angular density by means of the Pickands density and use it to construct our estimator. The estimator is asymptotically normal under certain regularity conditions. We also test it with simulated data and give an application to a real hydrological data set. Finally, we show that our estimator cannot be transferred directly to higher dimensions.     

Parameter estimation for generalized Pareto distribution by generalized probability weighted moment-equations

The generalized Pareto distribution (GPD) has been widely used to model exceedances over a threshold. This article generalizes the method of generalized probability weighted moments, and applies this method to estimate the parameters of GPD. The estimator is computationally easy. Some asymptotic results of this method are provided. Two simulations are carried out to investigate the behavior of this method and to compare them with other methods suggested in the literature. The simulation results show that the performance of the proposed method is better than some other methods. Finally, this method is applied to analyze a real-life data.     

Generalized inferential procedures for generalized Lorenz curves under the Pareto distribution

This paper considers problems of interval estimation and hypotheses testing for the generalized Lorenz curve under the Pareto distribution. Our approach is based on the concepts of generalized test variables and generalized pivotal quantities. The merits of the proposed procedures are numerically carried out and compared with asymptotic and bootstrap methods. Empirical evidence shows that the coverage accuracy of the proposed confidence intervals and the type I error control of the proposed exact tests are satisfactory. For illustration purposes, a real data set on median income of the 20 occupations in the United States Census of Population is analysed.     

A default Bayesian procedure for the generalized Pareto distribution

The generalized Pareto distribution is used to model the exceedances over a threshold in a number of fields, including the analysis of environmental extreme events and financial data analysis. We use this model in a default Bayesian framework where no prior information is available on unknown model parameters. Using a large simulation study, we compare the performance of our posterior estimations of parameters with other methods proposed in the literature. We show that our procedure also allows to make inferences in other quantities of interest in extreme value analysis without asymptotic arguments. We apply the proposed methodology to a real data set.     

A matching prior for extreme quantile estimation of the generalized Pareto distribution

Extreme quantile estimation plays an important role in risk management and environmental statistics among other applications. A popular method is the peaks-over-threshold (POT) model that approximate the distribution of excesses over a high threshold through generalized Pareto distribution (GPD). Motivated by a practical financial risk management problem, we look for an appropriate prior choice for Bayesian estimation of the GPD parameters that results in better quantile estimation. Specifically, we propose a noninformative matching prior for the parameters of a GPD so that a specific quantile of the Bayesian predictive distribution matches the true quantile in the sense of Datta et al. (2000).     

Parameter estimation of the generalized Pareto distribution—Part II

This is the second part of a paper which focuses on reviewing methods for estimating the parameters of the generalized Pareto distribution (GPD). The GPD is a very important distribution in the extreme value context. It is commonly used for modeling the observations that exceed very high thresholds. The ultimate success of the GPD in applications evidently depends on the parameter estimation process. Quite a few methods exist in the literature for estimating the GPD parameters. Estimation procedures, such as the maximum likelihood (ML), the method of moments (MOM) and the probability weighted moments (PWM) method were described in Part I of the paper. We shall continue to review methods for estimating the GPD parameters, in particular methods that are robust and procedures that use the Bayesian methodology. As in Part I, we shall focus on those that are relatively simple and straightforward to be applied to real world data.     

On von Mises type conditions for p-max stable laws, rates of convergence and generalized log Pareto distributions

In this article we obtain an alternative formulation of the von Mises type conditions for p-max stable laws in terms of generalized log Pareto distributions (glogPds). Relationship between the rate of convergence of extremes and the remainder terms in the von Mises type conditions is investigated. It is shown that the rate of convergence in the von Mises type conditions for p-max stable laws determines the distance of the underlying distribution function from a glogPd.     

Shared frailty models with baseline generalized Pareto distribution

Abstract

In this article, we have considered three different shared frailty models under the assumption of generalized Pareto Distribution as baseline distribution. Frailty models have been used in the survival analysis to account for the unobserved heterogeneity in an individual risks to disease and death. These three frailty models are with gamma frailty, inverse Gaussian frailty and positive stable frailty. Then we introduce the Bayesian estimation procedure using Markov chain Monte Carlo (MCMC) technique to estimate the parameters. We applied these three models to a kidney infection data and find the best fitted model for kidney infection data. We present a simulation study to compare true value of the parameters with the estimated values. Model comparison is made using Bayesian model selection criterion and a well-fitted model is suggested for the kidney infection data.     

Optimally robust estimators in generalized Pareto models

In this paper, we study the robustness properties of several procedures for the joint estimation of shape and scale in a generalized Pareto model. The estimators that we primarily focus upon, most bias robust estimator (MBRE) and optimal MSE-robust estimator (OMSE), are one-step estimators distinguished as optimally robust in the shrinking neighbourhood setting; that is, they minimize the maximal bias, respectively, on such a specific neighbourhood, the maximal mean squared error (MSE). For their initialization, we propose a particular location–dispersion estimator, MedkMAD, which matches the population median and kMAD (an asymmetric variant of the median of absolute deviations) against the empirical counterparts. These optimally robust estimators are compared to the maximum-likelihood, skipped maximum-likelihood, Cramér–von-Mises minimum distance, method-of-medians, and Pickands estimators. To quantify their deviation from robust optimality, for each of these suboptimal estimators, we determine the finite-sample breakdown point and the influence function, as well as the statistical accuracy measured by asymptotic bias, variance, and MSE – all evaluated uniformly on shrinking neighbourhoods. These asymptotic findings are complemented by an extensive simulation study to assess the finite-sample behaviour of the considered procedures. The applicability of the procedures and their stability against outliers are illustrated for the Danish fire insurance data set from the package evir.     

Parameter estimation of the generalized Pareto distribution—Part I

The generalized Pareto distribution (GPD) has been widely used in the extreme value framework. The success of the GPD when applied to real data sets depends substantially on the parameter estimation process. Several methods exist in the literature for estimating the GPD parameters. Mostly, the estimation is performed by maximum likelihood (ML). Alternatively, the probability weighted moments (PWM) and the method of moments (MOM) are often used, especially when the sample sizes are small. Although these three approaches are the most common and quite useful in many situations, their extensive use is also due to the lack of knowledge about other estimation methods. Actually, many other methods, besides the ones mentioned above, exist in the extreme value and hydrological literatures and as such are not widely known to practitioners in other areas. This paper is the first one of two papers that aim to fill in this gap. We shall extensively review some of the methods used for estimating the GPD parameters, focusing on those that can be applied in practical situations in a quite simple and straightforward manner.     

Bias-corrected maximum likelihood estimation of the parameters of the generalized Pareto distribution

ABSTRACT

We derive analytic expressions for the biases, to O(n^?1), of the maximum likelihood estimators of the parameters of the generalized Pareto distribution. Using these expressions to bias-correct the estimators in a selective manner is found to be extremely effective in terms of bias reduction, and can also result in a small reduction in relative mean squared error (MSE). In terms of remaining relative bias, the analytic bias-corrected estimators are somewhat less effective than their counterparts obtained by using a parametric bootstrap bias correction. However, the analytic correction out-performs the bootstrap correction in terms of remaining %MSE. It also performs credibly relative to other recently proposed estimators for this distribution. Taking into account the relative computational costs, this leads us to recommend the selective use of the analytic bias adjustment for most practical situations.     

A Note on the Complementary Mixture Pareto II Distribution

We introduce a new survival distribution, of Pareto type, that arises from a cure-mixture frailty model. We describe its properties and demonstrate connections with familiar distributions including the Pareto and exponential. We derive its characteristic function and moments.     

Entropy-based goodness-of-fit tests for the Pareto I distribution

In this article, having observed the generalized order statistics in a sample, we construct a test for the hypothesis that the underlying distribution is the Pareto I distribution. The Shannon entropy of generalized order statistics is used to test the null hypothesis.     

On a characteristic property of generalized Pareto distributions, extreme value distributions and their max domains of attraction

Using the independence of an arbitrary random variable Y and the weighted minima of independent, identically distributed random variables with weights depending on Y, we characterize extreme value distributions and generalized Pareto distributions. A discussion is made about an analogous characterization for distributions in the max domains of attraction of extreme value limit laws.     

On generalized order statistics from generalized pareto distribution

The aim of this paper is to estimate parameters of generalized Pareto distribution based on generalized order statistics. Some non-Bayesian methods, such as MLE, bootstrap and unbiased estimators have been obtained to develop point and interval estimations. Bayesian estimations have also been derived under LSE and LINEX loss functions. To compare the performances of the employed methods, numerical results have been computed. To illustrate dependence and association properties of generalized order statistics, correlation coefficient and some informational measures in closed form have been obtained.     

Linear Estimators and Predictors Based on Generalized Order Statistics from Generalized Pareto Distributions

Linear estimation and prediction based on several samples of generalized order statistics from generalized Pareto distributions is considered. Representations of best linear unbiased estimators (BLUEs) and best linear equivariant estimators in location-scale families are derived, as well as corresponding optimal linear predictors. Moreover, we study positivity of the linear estimators of the scale parameter. An example illustrates that the BLUE may attain negative values with positive probability in certain situations.     

Estimation for the Generalized Pareto Distribution Using Maximum Likelihood and Goodness of Fit

In this article, we introduce a new estimator for the generalized Pareto distribution, which is based on the maximum likelihood estimation and the goodness of fit. The asymptotic normality of the new estimator is shown and a small simulation. From the simulation, the performance of the new estimator is roughly comparable with maximum likelihood for positive values of the shape parameter and often much better than maximum likelihood for negative values.     

Multivariate Pareto Distributions: Inference and Financial Applications

Univariate Pareto distributions are extensively studied. In this article, we propose a Bayesian inference methodology in the context of multivariate Pareto distributions of the second kind (Mardia's type). Computational techniques organized around Gibbs sampling with data augmentation are proposed to implement Bayesian inference in practice. The new methods are shown to work well in artificial examples involving a trivariate distribution, and to an empirical application involving daily exchange rate data for four major currencies.     

Inferences on functions of Pareto parameters with application to income inequality measures

The Theil, Pietra, Éltetö and Frigyes measures of income inequality associated with the Pareto distribution function are expressed in terms of parameters defining the Pareto distribution. Inference procedures based on the generalized variable method, the large sample method, and the Bayesian method for testing of, and constructing confidence interval for, these measures are discussed. The results of Monte Carlo study are used to compare the performance of the suggested inference procedures from a population characterized by a Pareto distribution.     

Parametric Estimation Procedures in Multivariate Generalized Pareto Models

Abstract.  Modelling the tails of a multivariate distribution can be reasonably done by multivariate generalized Pareto distributions (GPDs). We present several methods of parametric estimation in these models, which use decompositions of the corresponding random vectors with the help of different versions of Pickands coordinates. The estimators are compared to each other with simulated data sets. To show the practical value of the methods, they are applied to a real hydrological data set.