On Asymptotic Normality of Hill's Estimator for the Exponent of Regular Variation

Haeusler, Erich; Teugels, Jozef L.

doi:10.1214/aos/1176349551

Cited by 191 publications

(104 citation statements)

References 12 publications

Supporting

Mentioning

102

Contrasting

Order By: Relevance

“…Condition (A.2) is the cornerstone in all proofs of asymptotic normality for extreme value estimators. It is used in [11] to prove the asymptotic normality of the Hill estimator and in [12] for one of its refinements.…”

Section: Asymptotic Resultsmentioning

confidence: 99%

A Hill Type Estimator of the Weibull Tail-Coefficient

Girard

2004

Communications in Statistics - Theory and Methods

View full text Add to dashboard Cite

We present a new estimator of the Weibull tail-coefficient. The Weibull tail-coefficient is defined as the regular variation coefficient of the inverse cumulative hazard function. Our estimator is based on the log-spacings of the upper order statistics. Therefore, it is very similar to the Hill estimator for the extreme value index. We prove the weak consistency and the asymptotic normality of our estimator. Its asymptotic as well as its finite sample performances are compared to classical ones.

show abstract

Section: Asymptotic Resultsmentioning

confidence: 99%

A Hill Type Estimator of the Weibull Tail-Coefficient

Girard

2004

Communications in Statistics - Theory and Methods

View full text Add to dashboard Cite

show abstract

“…setup under the additional assumption that k/ log log n → ∞. The asymptotic normality with a deterministic centring by 1/α requires additional assumptions on the distribution F of X and has been established in Haeusler and Teugels (1985); de Haan and ;Geluk, de Haan, S.I., and Starica (1997);Hill (2010). In this case,…”

Section: The Hill Methods For Tail Index Estimationmentioning

confidence: 99%

“…case, several authors investigated the asymptotic normality of the Hill estimator under various assumptions, including Davis and Resnick (1984), Beirlant and Teugels (1989) or Haeusler and Teugels (1985). In a primary work, Hsing (1991) showed a central limit theorem in a weak dependent setting under suitable mixing and stationary conditions.…”

Section: Introductionmentioning

confidence: 99%

On tail index estimation based on multivariate data

Dematteo

Clémençon

2015

Journal of Nonparametric Statistics

View full text Add to dashboard Cite

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, i.e. 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. Motivated by the statistical analysis of extremal spatial data in particular, we introduce the concept of (standard) heavy-tailed random field 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.

show abstract

“…As long as m n -1 is restricted based on properties of slowly varying L x , cf. Haeusler and Teugels (1985), in the iid case it is known (e.g. Hall, 1982) Resnick and Stȃricȃ (1995) and Hill (forthcoming, 2009, in press) and their citations.…”

Section: Tail Index Inferencementioning

confidence: 99%

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

Hill

2011

Journal of Statistical Planning and Inference

View full text Add to dashboard Cite

a b s t r a c tWe 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.

show abstract

On Asymptotic Normality of Hill's Estimator for the Exponent of Regular Variation

Cited by 191 publications

References 12 publications

A Hill Type Estimator of the Weibull Tail-Coefficient

A Hill Type Estimator of the Weibull Tail-Coefficient

On tail index estimation based on multivariate data

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

Contact Info

Product

Resources

About