PORT Hill and Moment Estimators for Heavy-Tailed Models

In this article, we use the peaks over random threshold (PORT)-methodology, and consider Hill and moment PORT-classes of extreme value index estimators. These classes of estimators are invariant not only to changes in scale, like the classical Hill and moment estimators, but also to changes in location. They are based on the sample of excesses over a random threshold, the order statistic X _[np]+1:n , 0 ≤ p < 1, being p a tuning parameter, which makes them highly flexible. Under convenient restrictions on the underlying model, these classes of estimators are consistent and asymptotically normal for adequate values of k, the number of top order statistics used in the semi-parametric estimation of the extreme value index γ. In practice, there may however appear a stability around a value distant from the target γ when the minimum is chosen for the random threshold, and attention is drawn for the danger of transforming the original data through the subtraction of the minimum. A new bias-corrected moment estimator is also introduced. The exact performance of the new extreme value index PORT-estimators is compared, through a large-scale Monte-Carlo simulation study, with the original Hill and moment estimators, the bias-corrected moment estimator, and one of the minimum-variance reduced-bias (MVRB) extreme value index estimators recently introduced in the literature. As an empirical example we estimate the tail index associated to a set of real data from the field of finance.     

A computational study of a quasi-PORT methodology for VaR based on second-order reduced-bias estimation

In this paper, we deal with the estimation, under a semi-parametric framework, of the Value-at-Risk (VaR) at a level p, the size of the loss occurred with a small probability p. Under such a context, the classical VaR estimators are the Weissman–Hill estimators, based on any intermediate number k of top-order statistics. But these VaR estimators do not enjoy the adequate linear property of quantiles, contrarily to the PORT VaR estimators, which depend on an extra tuning parameter q, with 0≤q<1. We shall here consider ‘quasi-PORT’ reduced-bias VaR estimators, for which such a linear property is obtained approximately. They are based on a partially shifted version of a minimum-variance reduced-bias (MVRB) estimator of the extreme value index (EVI), the primary parameter in Statistics of Extremes. Due to the stability on k of the MVRB EVI and associated VaR estimates, we propose the use of a heuristic stability criterion for the choice of k and q, providing applications of the methodology to simulated data and to log-returns of financial stocks.     

Reduced-Bias Location-Invariant Extreme Value Index Estimation: A Simulation Study

In this article, we deal with semi-parametric corrected-bias estimation of a positive extreme value index (EVI), the primary parameter in statistics of extremes. Under such a context, the classical EVI-estimators are the Hill estimators, based on any intermediate number k of top-order statistics. But these EVI-estimators are not location-invariant, contrarily to the PORT-Hill estimators, which depend on an extra tuning parameter q, with 0 ≤ q < 1, and where PORT stands for peaks over random threshold. On the basis of second-order minimum-variance reduced-bias (MVRB) EVI-estimators, we shall here consider PORT-MVRB EVI-estimators. Due to the stability on k of the MVRB EVI-estimates, we propose the use of a heuristic algorithm, for the adaptive choice of k and q, based on the bias pattern of the estimators as a function of k. Applications in the fields of insurance and finance will be provided.     

Semiparametric lower bounds for tail index estimation

We consider estimation of the tail index parameter from i.i.d. observations in Pareto and Weibull type models, using a local and asymptotic approach. The slowly varying function describing the non-tail behavior of the distribution is considered as an infinite dimensional nuisance parameter. Without further regularity conditions, we derive a local asymptotic normality (LAN) result for suitably chosen parametric submodels of the full semiparametric model. From this result, we immediately obtain the optimal rate of convergence of tail index parameter estimators for more specific models previously studied. On top of the optimal rate of convergence, our LAN result also gives the minimal limiting variance of estimators (regular for our parametric model) through the convolution theorem. We show that the classical Hill estimator is regular for the submodels introduced with limiting variance equal to the induced convolution theorem bound. We also discuss the Weibull model in this respect.     

On tail index estimation based on multivariate data

This article is devoted to the study of tail index estimation based on i.i.d. multivariate observations, drawn from a standard heavy-tailed distribution, that is, of which Pareto-like marginals share the same tail index. A multivariate central limit theorem for a random vector, whose components correspond to (possibly dependent) Hill estimators of the common tail index α, is established under mild conditions. We introduce the concept of (standard) heavy-tailed random vector of tail index α and show how this limit result can be used in order to build an estimator of α with small asymptotic mean squared error, through a proper convex linear combination of the coordinates. Beyond asymptotic results, simulation experiments illustrating the relevance of the approach promoted are also presented.     

Bias reduction for high quantiles

High quantile estimation is of importance in risk management. For a heavy-tailed distribution, estimating a high quantile is done via estimating the tail index. Reducing the bias in a tail index estimator can be achieved by using either the same order or a larger order of number of the upper order statistics in comparison with the theoretical optimal one in the classical tail index estimator. For the second approach, one can either estimate all parameters simultaneously or estimate the first and second order parameters separately. Recently, the first method and the second method via external estimators for the second order parameter have been applied to reduce the bias in high quantile estimation. Theoretically, the second method obviously gives rise to a smaller order of asymptotic mean squared error than the first one. In this paper we study the second method with simultaneous estimation of all parameters for reducing bias in high quantile estimation.     

Divergence based robust estimation of the tail index through an exponential regression model

The extreme value theory is very popular in applied sciences including finance, economics, hydrology and many other disciplines. In univariate extreme value theory, we model the data by a suitable distribution from the general max-domain of attraction characterized by its tail index; there are three broad classes of tails—the Pareto type, the Weibull type and the Gumbel type. The simplest and most common estimator of the tail index is the Hill estimator that works only for Pareto type tails and has a high bias; it is also highly non-robust in presence of outliers with respect to the assumed model. There have been some recent attempts to produce asymptotically unbiased or robust alternative to the Hill estimator; however all the robust alternatives work for any one type of tail. This paper proposes a new general estimator of the tail index that is both robust and has smaller bias under all the three tail types compared to the existing robust estimators. This essentially produces a robust generalization of the estimator proposed by Matthys and Beirlant (Stat Sin 13:853–880, 2003) under the same model approximation through a suitable exponential regression framework using the density power divergence. The robustness properties of the estimator are derived in the paper along with an extensive simulation study. A method for bias correction is also proposed with application to some real data examples.     

Does bias reduction with external estimator of second order parameter work for endpoint?

Bias reduction estimation for tail index has been studied in the literature. One method is to reduce bias with an external estimator of the second order regular variation parameter; see Gomes and Martins [2002. Asymptotically unbiased estimators of the tail index based on external estimation of the second order parameter. Extremes 5(1), 5–31]. It is known that negative extreme value index implies that the underlying distribution has a finite right endpoint. As far as we know, there exists no bias reduction estimator for the endpoint of a distribution. In this paper, we study the bias reduction method with an external estimator of the second order parameter for both the negative extreme value index and endpoint simultaneously. Surprisingly, we find that this bias reduction method for negative extreme value index requires a larger order of sample fraction than that for positive extreme value index. This finding implies that this bias reduction method for endpoint is less attractive than that for positive extreme value index. Nevertheless, our simulation study prefers the proposed bias reduction estimators to the biased estimators in Hall [1982. On estimating the endpoint of a distribution. Ann. Statist. 10, 556–568].     

A bias-reduced estimator for the mean of a heavy-tailed distribution with an infinite second moment

We use bias-reduced estimators of high quantiles of heavy-tailed distributions, to introduce a new estimator for the mean in the case of infinite second moment. The asymptotic normality of the proposed estimator is established and checked in a simulation study, by four of the most popular goodness-of-fit tests. The accuracy of the resulting confidence intervals is evaluated as well. We also investigate the finite sample behavior and compare our estimator with some versions of Peng's estimator of the mean (namely those based on Hill, t-Hill and Huisman et al. extreme value index estimators). Moreover, we discuss the robustness of the tail index estimators used in this paper. Finally, our estimation procedure is applied to the well-known Danish fire insurance claims data set, to provide confidence bounds for the means of weekly and monthly maximum losses over a period of 10 years.     

Local Estimation of the Second-Order Parameter in Extreme Value Statistics and Local Unbiased Estimation of the Tail Index

We develop and study in the framework of Pareto-type distributions a class of nonparametric kernel estimators for the conditional second order tail parameter. The estimators are obtained by local estimation of the conditional second order parameter using a moving window approach. Asymptotic normality of the proposed class of kernel estimators is proven under some suitable conditions on the kernel function and the conditional tail quantile function. The nonparametric estimators for the second order parameter are subsequently used to obtain a class of bias-corrected kernel estimators for the conditional tail index. In particular it is shown how for a given kernel function one obtains a bias-corrected kernel function, and that replacing the second order parameter in the latter with a consistent estimator does not change the limiting distribution of the bias-corrected estimator for the conditional tail index. The finite sample behavior of some specific estimators is illustrated with a simulation experiment. The developed methodology is also illustrated on fire insurance claim data.     

Moment-based tail index estimation

A general method of tail index estimation for heavy-tailed time series, based on examining the growth rate of the logged sample second moment of the data was proposed and studied in Meerschaert and Scheffler (1998. A simple robust estimator for the thickness of heavy tails. J. Statist. Plann. Inference 71, 19–34) as well as Politis (2002. A new approach on estimation of the tail index. C. R. Acad. Sci. Paris, Ser. I 335, 279–282). To improve upon the basic estimator, we introduce a scale-invariant estimator that is computed over subsets of the whole data set. We show that the new estimator, under some stronger conditions on the data, has a polynomial rate of consistency for the tail index. Empirical studies explore how the new method compares with the Hill, Pickands, and DEdH estimators.     

Weibull tail-distributions revisited: A new look at some tail estimators

In this paper, we propose to include Weibull tail-distributions in a more general family of distributions. In particular, the considered model also encompasses the whole Fréchet maximum domain of attraction as well as log-Weibull tail-distributions. The asymptotic normality of some tail estimators based on the log-spacings between the largest order statistics is established in a unified way within the considered family. This result permits to understand the similarity between most estimators of the Weibull tail-coefficient and the Hill estimator. Some different asymptotic properties, in terms of bias, rate of convergence, are also highlighted.     

On Smooth Statistical Tail Functionals

Many estimators of the extreme value index of a distribution function F that are based on a certain number k _n of largest order statistics can be represented as a statistical tail function al, that is a functional T applied to the empirical tail quantile function Q _n. We study the asymptotic behaviour of such estimators with a scale and location invariant functional T under weak second order conditions on F . For that purpose first a new approximation of the empirical tail quantile function is established. As a consequence we obtain weak consistency and asymptotic normality of T ( Q _n) if T is continuous and Hadamard differentiable, respectively, at the upper quantile function of a generalized Pareto distribution and k _pn tends to infinity sufficiently slowly. Then we investigate the asymptotic variance and bias. In particular, those functionals T re characterized that lead to an estimator with minimal asymptotic variance. Finally, we introduce a method to construct estimators of the extreme value index with a made-to-order asymptotic behaviour     

A robust prediction error criterion for pareto modelling of upper tails

Estimation of the Pareto tail index from extreme order statistics is an important problem in many settings. The upper tail of the distribution, where data are sparse, is typically fitted with a model, such as the Pareto model, from which quantities such as probabilities associated with extreme events are deduced. The success of this procedure relies heavily not only on the choice of the estimator for the Pareto tail index but also on the procedure used to determine the number k of extreme order statistics that are used for the estimation. The authors develop a robust prediction error criterion for choosing k and estimating the Pareto index. A Monte Carlo study shows the good performance of the new estimator and the analysis of real data sets illustrates that a robust procedure for selection, and not just for estimation, is needed.     

A method of moments estimator of tail dependence in meta-elliptical models

A meta-elliptical model is a distribution function whose copula is that of an elliptical distribution. The tail dependence function in such a bivariate model has a parametric representation with two parameters: a tail parameter and a correlation parameter. The correlation parameter can be estimated by robust methods based on the whole sample. Using the estimated correlation parameter as plug-in estimator, we then estimate the tail parameter applying a modification of the method of moments approach proposed in the paper by Einmahl et al. (2008). We show that such an estimator is consistent and asymptotically normal. Further, we derive the joint limit distribution of the estimators of the two parameters. We illustrate the small sample behavior of the estimator of the tail parameter by a simulation study and on real data, and we compare its performance to that of the competitive estimators.     

On moments of dual generalized order statistics from Topp-Leone distribution

In this paper, we have derived exact and explicit expressions for the ratio and inverse moments of dual generalized order statistics from Topp-Leone distribution. This result includes the single and product moments of order statistics and lower records . Further, based on n dual generalized order statistics, we have deduced the expression for Maximum likelihood estimator (MLE) and Uniformly minimum variance unbiased estimator (UMVUE) for the shape parameter of Topp-Leone distribution. Finally, based on order statistics and lower records, a simulation study is being carried out to check the efficiency of these estimators.     

α-stable laws for noncoding regions in DNA sequences

In this work, we analyze the long-range dependence parameter for a nucleotide sequence in several different transformations. The long-range dependence parameter is estimated by the approximated maximum likelihood method, by a novel estimator based on the spectral envelope theory, by a regression method based on the periodogram function, and also by the detrended fluctuation analysis method. We study the length distribution of coding and noncoding regions for all Homo sapiens chromosomes available from the European Bioinformatics Institute. The parameter of the tail rate decay is estimated by the Hill estimator ?α. We show that the tail rate decay is greater than 2 for coding regions, while for almost all noncoding regions it is less than 2.     

Hill estimator of projections of functional data on principal components

Functional principal component scores are commonly used to reduce mathematically infinitely dimensional functional data to finite dimensional vectors. In certain applications, most notably in finance, these scores exhibit tail behaviour consistent with the assumption of regular variation. Knowledge of the index of the regular variation, α, is needed to apply methods of extreme value theory. The most commonly used method of the estimation of α is the Hill estimator. We derive conditions under which the Hill estimator computed from the sample scores is consistent for the tail index of the unobservable population scores.     

Tail index estimation in small smaples Simulation results for independent and ARCH-type financial return models

Estimation of the tail index of stationary, fat-tailed return distributions is non-trivial since the well-known Hill estimator is optimal only under iid draws from an exact Pareto model. We provide a small sample simulation study of recently suggested adaptive estimators under ARCH-type dependence. The Hill estimator’s performance is found to be dominated by a ratio estimator. Dependence increases estimation error which can remain substantial even in larger data sets. As small sample bias is related to the magnitude of the tail index, recent standard applications may have overestimated (underestimated) the risk of assets with low (high) degrees of fat-tailedness. This paper is a shortened version of the Berkeley Research Program in Finance Working Paper RPF-295. Thanks are to the Center for Mathematical Sciences at Munich University of Technology for generously providing access to computer facilities and to participants at the IAFE 2001 Budapest, OR 2002 Klagenfurt, EIR 2002 London, DGF 2002 Cologne, FBI 2002 Karlsruhe conferences and the 2001 Wallis Workshop for helpful comments. Two anonymous referees provided helpful suggestions in streamlining the material. Niklas Wagner acknowledges a Maple program by Klaus Kiefersbeck and financial support by Deutsche Forschungsgemeinschaft (DFG).