Estimation Of Tail Index Research Articles

Asymptotic expansions for infinite weighted convolutions of heavy tail distributions and applications

The authors establish some asymptotic expansions for infinite weighted convolution of distributions having regularly varying tails. Applications to linear time series models, tail index estimation, compound sums, queueing theory, branching processes, infinitely divisible distributions and implicit transient renewal equations are given.A noteworthy feature of the approach taken in this paper is that through the introduction of objects, which the authors call the Laplace characters, a link is established between tail area expansions and algebra. By virtue of this representation approach, a unified method to establish expansions across a variety of problems is presented and, moreover, the method can be easily programmed so that a computer algebra package makes implementation of the method not only feasible but simple.

Extreme Volatilities, Financial Crises and L-Moment Estimations of Tail Indexes

Following Bali and Weinbaum (2005) and Maillet et al. (2010), we present several estimates of volatilities computed with high- and low frequency data and complement their results using additional measures of risk and several alternative methods for Tail-index estimation. The aim here is to confirm previous results regarding the slope of the tail of various risk measure distributions, in order to define the high watermarks of market risks. We also produce synthetic general results concerning the method of estimation of the Tail-indexes related to expressions of the L-moments. Based on estimates of Tail-indexes, retrieved from the high frequency 30� sampled CAC40 French stock Index series from the period 1997-2009, using Non-parametric Generalized Hill, Maximum Likelihood and various kinds of L-moment Methods for the estimation of both a Generalized Extreme Value density and a Generalized Pareto Distribution, we confirm that a heavy-tail density specification of the Log-volatility is not necessary.

On functional central limit theorems for dependent, heterogeneous arrays with applications to tail index and tail dependence estimation

We establish invariance principles for a large class of dependent, heterogeneous arrays. The theory equally covers conventional arrays, and inherently degenerate tail arrays popularly encountered in the extreme value theory literature including sample means and covariances of tail events and exceedances. For tail arrays we trim dependence assumptions down to a minimum leaving non-extremes and joint distributions unrestricted, covering geometrically ergodic, mixing, and mixingale processes, in particular linear and nonlinear distributed lags with long or short memory, linear and nonlinear GARCH, and stochastic volatility. Of practical importance the limit theory can be used to characterize the functional limit distributions of a tail index estimator, the tail quantile process, and a bivariate extremal dependence estimator under substantially general conditions.

A note on the asymptotic variance at optimal levels of a bias-corrected Hill estimator

For heavy tails, with a positive tail index γ , classical tail index estimators, like the Hill estimator, are known to be quite sensitive to the number of top order statistics k used in the estimation, whereas second-order reduced-bias estimators show much less sensitivity to changes in k . In the recent minimum-variance reduced-bias (MVRB) tail index estimators, the estimation of the second order parameters in the bias has been performed at a level k 1 of a larger order than that of the level k at which we compute the tail index estimators. Such a procedure enables us to keep the asymptotic variance of the new estimators equal to the asymptotic variance of the Hill estimator, for all k at which we can guarantee the asymptotic normality of the Hill statistics. These values of k , as well as larger values of k , will also enable us to guarantee the asymptotic normality of the reduced-bias estimators, but, to reach the minimal mean squared error of these MVRB estimators, we need to work with levels k and k 1 of the same order. In this note we derive the way the asymptotic variance varies as a function of q , the finite limiting value of k / k 1 , as the sample size n increases to infinity.

Tail index estimation for heavy tails: accommodation of bias in the excesses over a high threshold

In statistics of extremes, inference is often based on the excesses over a high random threshold. Those excesses are approximately distributed as the set of order statistics associated to a sample from a generalized Pareto model. We then get the so-called “maximum likelihood” estimators of the tail index γ. In this paper, we are interested in the derivation of the asymptotic distributional properties of a similar “maximum likelihood” estimator of a positive tail index γ, based also on the excesses over a high random threshold, but with a trial of accommodation of bias in the Pareto model underlying those excesses. We next proceed to an asymptotic comparison of the two estimators at their optimal levels. An illustration of the finite sample behaviour of the estimators is provided through a small-scale Monte Carlo simulation study.

Weak Convergence of the Empirical Mean Excess Process with Application to Estimate the Negative Tail Index

Let Y i , 1 ≤ i ≤ n be i.i.d. random variables with the generalized Pareto distribution W γ,σ with γ < 0. We define the empirical mean excess process with respect to {Y i , 1 ≤ i ≤ n} as in Eq. 2.1 (see below) and investigate its weak convergence. As an application, two new estimators of the negative tail index γ are constructed based on the linear regression to the empirical mean excess function and their consistency and asymptotic normality are obtained.

A moving window approach for nonparametric estimation of the conditional tail index

We present a nonparametric family of estimators for the tail index of a Pareto-type distribution when covariate information is available. Our estimators are based on a weighted sum of the log-spacings between some selected observations. This selection is achieved through a moving window approach on the covariate domain and a random threshold on the variable of interest. Asymptotic normality is proved under mild regularity conditions and illustrated for some weight functions. Finite sample performances are presented on a real data study.

Tail Index Estimation for Heavy-Tailed Models: Accommodation of Bias in Weighted Log-Excesses

Summary We are interested in the derivation of the distributional properties of a weighted log-excesses estimator of a positive tail index γ. One of the main objectives of such an estimator is the accommodation of the dominant component of asymptotic bias, together with the maintenance of the asymptotic variance of the maximum likelihood estimator of γ, under a strict Pareto model. We consider the external estimation not only of a second-order shape parameter ρ but also of a second-order scale parameter β. This will enable us to reduce the asymptotic variance of the final estimators under consideration, compared with second-order reduced bias estimators that are already available in the literature. The second-order reduced bias estimators that are considered are also studied for finite samples, through Monte Carlo techniques, as well as applied to real data in the field of finance.

Improved reduced-bias tail index and quantile estimators

In this paper, we deal with bias reduction techniques for heavy tails, trying to improve mainly upon the performance of classical high quantile estimators. High quantiles depend strongly on the tail index γ , for which new classes of reduced-bias estimators have recently been introduced, where the second-order parameters in the bias are estimated at a level k 1 of a larger order than the level k at which the tail index is estimated. Doing this, it was seen that the asymptotic variance of the new estimators could be kept equal to the one of the popular Hill estimators. In a similar way, we now introduce new classes of tail index and associated high quantile estimators, with an asymptotic mean squared error smaller than that of the classical ones for all k in a large class of heavy-tailed models. We derive their asymptotic distributional properties and compare them with those of alternative estimators. Next to that, an illustration of the finite sample behavior of the estimators is also provided through a Monte Carlo simulation study and the application to a set of real data in the field of insurance.

On estimation of the exponent of regular variation using a sample with missing observations

Let ( X n ) be a sequence of possibly dependent random variables with the same marginal distribution function F, such that 1 - F ( x ) = x - α L ( x ) , α > 0 , where L ( x ) is a slowly varying function. In this paper the Hill estimator of the exponent of regular variation based on a sample with missing observations from the sequence ( X n ) is considered. The asymptotic consistency was proved under some general conditions. This extends results of Hsing [1991. On tail index estimation using dependent data. Ann. Statist. 19, 1547–1569].

Computer-intensive rate estimation, diverging statistics and scanning

A general rate estimation method is proposed that is based on studying the in-sample evolution of appropriately chosen diverging/converging statistics. The proposed rate estimators are based on simple least squares arguments, and are shown to be accurate in a very general setting without requiring the choice of a tuning parameter. The notion of scanning is introduced with the purpose of extracting useful subsamples of the data series; the proposed rate estimation method is applied to different scans, and the resulting estimators are then combined to improve accuracy. Applications to heavy tail index estimation as well as to the problem of estimating the long memory parameter are discussed; a small simulation study complements our theoretical results.

Smooth tail-index estimation

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.

A simple second-order reduced bias’ tail index estimator

In this article, we are interested in the direct estimation of the dominant component of the bias of a classical tail index estimator, such as the Hill estimator, used here for illustration of the procedure. Such an estimated bias is then directly removed from the original estimator. The second-order parameters in the bias are based on a number of top order statistics, larger than the one we should use for the estimation of the tail index γ, so that there is no change in the asymptotic variance of the new reduced bias’ tail index estimator, which is kept equal to the asymptotic variance of the classical original one, contrarily to what happens with most of the reduced bias’ estimators available in the literature. The asymptotic distributional behaviour of the proposed estimators of γ is derived, under a second-order framework, and their finite sample properties are also obtained through Monte Carlo simulation techniques.

Reduced‐bias tail index estimation and the jackknife methodology

In the context of regularly varying tails, we first analyze a generalization of the classical Hill estimator of a positive tail index, with members that are not asymptotically more efficient than the original one. This has led us to propose alternative classical tail index estimators, that may perform asymptotically better than the Hill estimator. As the improvement is not really significant, we also propose generalized jackknife estimators based on any two members of these two classes. These generalized jackknife estimators are compared with the Hill estimator and other reduced‐bias estimators available in the literature, asymptotically, and for finite samples, through the use of Monte Carlo simulation. The finite‐sample behaviour of the new reduced‐bias estimators is also illustrated through a practical example in the field of finance.

A Sturdy Reduced-Bias Extreme Quantile (VaR) Estimator

Please see the supplementary material for an important correction.The main objective of statistics of extremes lies in the estimation of quantities related to extreme events. In many areas of application, such as statistical quality control, insurance, and finance, a typical requirement is to estimate a high quantile, that is, the value at risk at a level p (VaRp), high enough so that the chance of an exceedance of that value is equal to p, small. In this article we deal with the semiparametric estimation of VaRp for heavy tails. The classical semiparametric estimators of parameters characterizing the tail behavior of the underlying model F usually exhibit a high bias for low thresholds, that is, for large values of k, the number of top order statistics used for the estimation. We shall here deal with bias reduction techniques for heavy tails, trying to improve the performance of the classical high quantile estimators through the use of an adequate bias-corrected tail index estimator. The new high quantile estimators have a mean squared error smaller than that of the classical estimators, even for small values of k. They are, thus, alternatives to the classical estimators not only around optimal levels but also for other levels. The asymptotic distributional properties of the proposed classes of estimators are derived. The estimators are compared with alternative ones, not only asymptotically but also for finite samples, through Monte Carlo techniques. An application to the analysis of different datasets in the field of finance is also provided.

A new class of estimators of a “scale” second order parameter

For a large class of heavy-tailed distribution functions F in the domain of attraction for maxima of an Extreme Value distribution with tail index γ>0, the function A(t), controlling the speed of convergence of maximum values, linearly normalized, towards a non-degenerate limiting random variable, may be parameterized as \(A(t)=\gamma\ \beta\ t^\rho\), ρ < 0, β∈ℝ, where β and ρ are second order parameters. The estimation of ρ, the “shape” second order parameter has been extensively addressed in the literature, but practically nothing has been done related to the estimation of the “scale” second order parameter β. In this paper, and motivated by the importance of a reliable β-estimation in recent reduced bias tail index estimators, we shall deal with such a topic. Under a semi-parametric framework, we introduce a class of β-estimators and study their consistency. We deal with the conditions enabling us to get the asymptotic normality of the members of this class, and we illustrate the behaviour of the estimators, through Monte Carlo simulation techniques.

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.

Testing for small bias of tail index estimators

We determine the joint asymptotic normality of kernel and weighted least-squares estimators of the upper tail index of a regularly varying distribution when each estimator is a bivariate function of two parameters: the tuning parameter is motivated by possible underlying second-order behavior in regular variation, while no such behavior is assumed, and the fraction parameter determines that upper portion of the sample on which the estimator is based. Under the hypothesis that the scaled asymptotic biases of the estimators vanish uniformly in the parameter points considered, these results imply joint asymptotic normality for deviations of ratios of the estimators from 1, which in turn yield asymptotic chi-square tests for checking the small-bias hypothesis, equivalent to the constructibility of asymptotic confidence intervals. The test procedure suggests adaptive choices of the tuning and fraction parameters: data-driven (t)estimators.

Tail Index Estimation in Models of Generalized Order Statistics

ABSTRACT This article is concerned with generalized order statistics based on a distribution function F which belongs to the domain of attraction of the Fréchet distribution G α, α > 0. We consider the Hill estimator of α which is based on extreme and intermediate generalized order statistics. Under suitable conditions its consistency and asymptotic normality is shown.

Autoregressive Conditional Tail Behavior and Results on Government Bond Yield Spreads

Previous evidence in empirical finance indicates the potential usefulness of modeling time-variation particularly in the tails of speculative return distributions. Based on results from extreme value theory, the present paper proposes a fixed changepoint Pareto-type autoregressive conditional tail model. Regression-based parameter estimation of the unobservable time-varying tail index is carried out via classical Kalman filtering. A model application highlights the tail index dynamics for daily changes in Government bond yield spreads between the US-Dollar and the Euro zone.