M. Ivette Gomes

The bootstrap methodology in statistics of extremes: Choice of the optimal sample fraction

The main objective of statistics of extremes is the prediction of rare events, and its primary problem has been the estimation of the tail index γ, usually performed on the basis of the largest k order statistics in the sample or on the excesses over a high level u. The question that has been often addressed in practical applications of extreme value theory is the choice of either k or u, and an adaptive estimation of γ. We shall be here mainly interested in the use of the bootstrap methodology to estimate γ adaptively, and although the methods provided may be applied, with adequate modifications, to the general domain of attraction of Gγ, γ ∈ ℝ, we shall here illustrate the methods for heavy right tails, i.e. for γ > 0. Special relevance will be given to the use of an auxiliary statistic that is merely the difference of two estimators with the same functional form as the estimator under study, computed at two different levels. We shall also compare, through Monte Carlo simulation, these bootstrap methodologies with other data-driven choices of the optimal sample fraction available in the literature.

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.

Semi-parametric Estimation of the Second Order Parameter in Statistics of Extremes

We present a class of semi-parametric estimators for the second order parameter related to a probability distribution with a regularly varying tail. The second order parameter plays an important role whenever dealing with optimization problems in statistics of extreme values. Consistency and asymptotic normality are proven under appropriate conditions.

Generalizations of the Hill estimator – asymptotic versus finite sample behaviour☆

Abstract The main goal of this paper is to present generalized Hill estimators parametrized in a positive real α (and equal to the Hill estimator when α =1), which are asymptotically more efficient than the Hill estimator for a large region of values of α for any point of the ( γ , ρ )-plane, where γ >0 is the tail index , related to the heaviness of the tail 1− F of the underlying model F , and ρ ⩽0 is the second-order parameter , related to the rate of convergence of maximum values, linearly normalized, towards its limit. The practical validation of asymptotic results for small finite samples is done by means of simulation techniques in Frechet and Burr models, and some indications are provided on the choice of α .

Penultimate limiting forms in extreme value theory

Let {X n}n≧1 be a sequence of independent, identically distributed random variables. If the distribution function (d.f.) ofM n=max (X 1,…,X n), suitably normalized with attraction coefficients {αn}n≧1(αn>0) and {b n}n≧1, converges to a non-degenerate d.f.G(x), asn→∞, it is of interest to study the rate of convergence to that limit law and if the convergence is slow, to find other d.f.s which better approximate the d.f. of(M n−bn)/an thanG(x), for moderaten. We thus consider differences of the formF n(anx+bn)−G(x), whereG(x) is a type I d.f. of largest values, i.e.,G(x)≡Λ(x)=exp (-exp(−x)), and show that for a broad class of d.f.sF in the domain of attraction of Λ, there is a penultimate form of approximation which is a type II [Ф α(x)=exp (−x−α), x>0] or a type III [Ψ α(x)= exp (−(−x)α), x<0] d.f. of largest values, much closer toF n(anx+bn) than the ultimate itself.

Reduced-bias tail index estimators under a third order framework

In this article, we are interested in the comparison, under a third-order framework, of classes of second-order, reduced-bias tail index estimators, giving particular emphasis to minimum-variance reduced-bias estimators of the tail index γ. The full asymptotic distributional properties of the proposed classes are derived under a third-order framework and the estimators are compared with other alternatives, not only asymptotically, but also for finite samples through Monte Carlo techniques. An application to the log-exchange rates of the Euro against the USA Dollar is also provided.

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.

Bias reduction and explicit semi-parametric estimation of the tail index☆

In this paper, and in a context of regularly varying tails, we analyse particular but interesting cases of the maximum likelihood and least squares estimators proposed by Feuerverger and Hall (Ann. Statist. 27 (1999) 760). All these estimators are alternatives to a well-known estimator of the tail index, the Hill estimator (Ann. Statist. 3 (1997) 1163), and jointly with the generalized jackknife estimators in Gomes et al. (Extremes 2 (2000) 207, Portug. Math. 59 (2002) 393) have essentially in mind a reduction in bias, preferably without increasing mean squared error, leading to semi-parametric estimators of the tail index with a better performance than the classical estimators, provided we may use extreme-value data relatively deep into the sample.

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

A Class of Asymptotically Unbiased Semi-parametric Estimators of the Tail Index

exchange rates.