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 of a tail index estimator through an external estimation of the second-order parameter

In this paper, we first consider a class of consistent semi-parametric estimators of a positive tail index γ, parameterised in a tuning or control parameter α. Such a control parameter enables us to have access, for any available sample, to an estimator of the tail index γ with a null dominant component of asymptotic bias, and consequently with a reasonably flat mean squared error pattern, as a function of k, the number of top-order statistics considered. Such a control parameter depends on a second-order parameter ρ, which will be adequately estimated so that we may achieve a high efficiency relative to the classical Hill estimator, provided we use a number of top-order statistics larger than the one usually required for the estimation through the Hill estimator. An illustration of the behaviour of the estimators is provided, through the analysis of the daily log-returns on the Euro–US$ exchange rates.     

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.     

Additive Transformation Models for Multivariate Interval-Censored Data

ABSTRACT

We present a new estimator of extreme quantiles dedicated to Weibull tail distributions. This estimate is based on a consistent estimator of the Weibull tail coefficient. This parameter is defined as the regular variation coefficient of the inverse cumulative hazard function. We give conditions in order to obtain the weak consistency and the asymptotic distribution of the extreme quantiles estimator. Its asymptotic as well as its finite sample performances are compared to classical ones.     

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.     

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.     

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.     

Abelian and Tauberian Theorems on the Bias of the Hill Estimator

The bias of Hill's estimator for the positive extreme value index of a distribution is investigated in relation to the convergence rate in the regular variation property of the tail function of the common distribution of the sample and the corresponding tail quantile function. Based on the theory of generalized regular variation, natural second-order conditions are proposed which both imply and are implied by convergence of the expectation of Hill's estimator to the extreme value index at certain rates. A comparison with second-order conditions encountered in the literature is made.     

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.     

Extremal memory of stochastic volatility with an application to tail shape inference

We characterize joint tails and tail dependence for a class of stochastic volatility processes. We derive the exact joint tail shape of multivariate stochastic volatility with innovations that have a regularly varying distribution tail. This is used to give four new characterizations of tail dependence. In three cases tail dependence is a non-trivial function of linear volatility memory parametrically represented by tail scales, while tail power indices do not provide any relevant dependence information. Although tail dependence is associated with linear volatility memory, tail dependence itself is nonlinear. In the fourth case a linear function of tail events and exceedances is linearly independent. Tail dependence falls in a class that implies the celebrated Hill (1975) tail index estimator is asymptotically normal, while linear independence of nonlinear tail arrays ensures the asymptotic variance is the same as the iid case. We illustrate the latter finding by simulation.     

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.     

OPTIMAL SEMIPARAMETRIC INFERENCE FOR THE TAIL INDEX BASED ON RATIOS OF THE LARGEST EXTREMES

The k largest order statistics in a random sample from a common heavy‐tailed parent distribution with a regularly varying tail can be characterized as Fréchet extremes. This paper establishes that consecutive ratios of such Fréchet extremes are mutually independent and distributed as functions of beta random variables. The maximum likelihood estimator of the tail index based on these ratios is derived, and the exact distribution of the maximum likelihood estimator is determined for fixed k, and the asymptotic distribution as k →∞ . Inferential procedures based upon the maximum likelihood estimator are shown to be optimal. The Fréchet extremes are not directly observable, but a feasible version of the maximum likelihood estimator is equivalent to Hill's statistic. A simple diagnostic is presented that can be used to decide on the largest value of k for which an assumption of Fréchet extremes is sustainable. The results are illustrated using data on commercial insurance claims arising from fires and explosions, and from hurricanes.     

An Estimator of the Exponent of Regular Variation Based on K-Record Values

Let {X _n:n ≥ 1} be an i.i.d. sequence of random variables with a continuous distribution function F. Under the assumption that the upper tail of Fis regularly varying with exponent 1/α, α > 0, we study the asymptotic properties of an estimator of α based on k-record values.     

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.     

Local rates of convergence for consistent estimators of location of translation families

We consider the estimation of a location parameter θ in a one-sample problem. A measure of the asymptotic performance of an estimator sequence {T_n} = T is given by the exponential rate of convergence to zero of the tail probability, which for consistent estimator sequences is bounded by a constant, B (θ, ?), called the Bahadur bound. We consider two consistent estimators: the maximum-likelihood estimator (mle) and a consistent estimator based on a likelihood-ratio statistic, which we call the probability-ratio estimator (pre). In order to compare the local behaviour of these estimators, we obtain Taylor series expansions in ? for B (θ, ?) and the exponential rates of the mle and pre. Finally, some numerical work is presented in which we consider a variety of underlying distributions.     

Improved Confidence Statements for the Binomial Parameter

In the computation of two-sided confidence intervals for the binomial parameter p (using the binomial mass function), it is known that such intervals achieve a confidence coefficient that in general is not equal to the confidence level 1 – α, say. In this article we present some general results on the confidence coefficient and tabulate them for selected pairs (α, n = number of trials). We treat only the nominal equal tail probability case because it is the most commonly taught and used.     

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].     

Fat tail distributions and local thin tail alternatives

The behaviour of the Hill estimator for the tail index of fat tailed distributions in the presence of local alternatives which have a thin tail is investigated. The converse problem is also briefly addressed. A local thin tail alternative can severely bias the Hill statistic. The relevance of this issue for the class of stable distributions is discussed. We conduct a small simulation study to support the analysis. In the conclusion it is argued that for moderate out of sample quantile analysis the problem of local alternatives may be less pressing.     

Beyond tail median and conditional tail expectation: Extreme risk estimation using tail Lp-optimization

The conditional tail expectation (CTE) is an indicator of tail behavior that takes into account both the frequency and magnitude of a tail event. However, the asymptotic normality of its empirical estimator requires that the underlying distribution possess a finite variance; this can be a strong restriction in actuarial and financial applications. A valuable alternative is the median shortfall (MS), although it only gives information about the frequency of a tail event. We construct a class of tail L^p-medians encompassing the MS and CTE. For p in (1,2), a tail L^p-median depends on both the frequency and magnitude of tail events, and its empirical estimator is, within the range of the data, asymptotically normal under a condition weaker than a finite variance. We extrapolate this estimator and another technique to extreme levels using the heavy-tailed framework. The estimators are showcased on a simulation study and on real fire insurance data.