Asymptotic normality of location invariant heavy tail index estimator

Motivated by Fraga Alves (Extremes 4:199–217, 2001)’s work, a new class of location invariant Hill-type estimators for the tail index of a heavy tailed distribution is proposed in the paper. Its asymptotic behavior is derived, and the optimal choice of the sample fraction is discussed by mean squared error. Asymptotic comparisons and simulation studies are presented to show that the new estimator performs well compared to the known ones.     

An estimator of the tail index based on increment ratio statistics

In this paper, we introduce an increment ratio statistic (IR ^N,m ) based estimator for estimation of the tail index of a heavy-tailed distribution. For i.i.d. observations depending on the zone of attraction of an α-stable law (0 < α < 2), the IR ^N,m statistic converges to a decreasing function L(α) as both the sample size N and bandwidth parameter m tend to infinity. We obtain a rate of decay of the bias EIR ^N,m −L(α) and mean square error E(IR ^N,m −L(α))². A central limit theorem (IR ^N,m −EIR ^N,m )⟹ (0,σ²(α)) is also obtained. Monte Carlo simulations show that our tail index estimator has quite good empirical mean square error and, unlike the Hill estimator, is not so sensitive to a change of bandwidth parameter m. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-25/08.     

Estimation of the conditional tail index using a smoothed local Hill estimator

For heavy-tailed distributions, the so-called tail index is an important parameter that controls the behavior of the tail distribution and is thus of primary interest to estimate extreme quantiles. In this paper, the estimation of the tail index is considered in the presence of a finite-dimensional random covariate. Uniform weak consistency and asymptotic normality of the proposed estimator are established and some illustrations on simulations are provided.     

The harmonic moment tail index estimator: asymptotic distribution and robustness

Asymptotic properties of the harmonic moment tail index Estimator are derived for distributions with regularly varying tails. The estimator shows good robustness properties and stands out for its simplicity. A tuning parameter allows for regulating the trade-off between robustness and efficiency. Small sample properties are illustrated by a simulation study.     

Double-thresholded estimator of extreme value index

The purpose of this Note is to propose an estimator of the extreme value index constructed by using only the number of points exceeding random thresholds. We prove the weak consistency and the asymptotic normality of this estimator. We deduce from this last result that the rate of convergence of our estimator is in a power of the sample size. To our knowledge, this rate of convergence is not reached by any other estimate of the extreme value index. Through a simulation, we compare our estimator to the moment estimator (Dekkers et al., Ann. Statist. 17 (1989) 1833–1855). To cite this article: L. Gardes, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

A two-step estimator of the extreme value index

In this paper, we build a two-step estimator , which satisfies , where is the well-known maximum likelihood estimator of the extreme value index. Since the two-step estimator can be calculated easily as a function of the observations, it is much simpler to use in practice. By properly choosing the first step estimator, such as the Pickands estimator, we can even get a shift and scale invariant estimator with the above property. The author thanks Laurens de Haan for motivating this work and giving helpful comments. The author also thanks two anonymous referees for their useful comments.     

Testing the tail index in autoregressive models

We propose a class of nonparametric tests on the Pareto tail index of the innovation distribution in the linear autoregressive model. The simulation study illustrates a good performance of the tests. Such tests have various applications in a study of flood flows, rainflow data, behavior of solids, atmospheric ozone layer and reliability analysis, in communication engineering, in stock markets and insurance. Research of J. Jurečková and J. Picek was partly supported by Czech Republic Grant 201/05/2340, by the Research Project LC06024 and by the NSF grant DMS 0071619. Research of H. L. Koul was partly supported by the NSF grants DMS 0071619 and 0704130.     

An estimator of the stable tail dependence function based on the empirical beta copula

The replacement of indicator functions by integrated beta kernels in the definition of the empirical tail dependence function is shown to produce a smoothed version of the latter estimator with the same asymptotic distribution but superior finite-sample performance. The link of the new estimator with the empirical beta copula enables a simple but effective resampling scheme.     

Estimation of the tail index for lattice-valued sequences

If one applies the Hill, Pickands or Dekkers–Einmahl–de Haan estimators of the tail index of a distribution to data which are rounded off one often observes that these estimators oscillate strongly as a function of the number k of order statistics involved. We study this phenomenon in the case of a Pareto distribution. We provide formulas for the expected value and variance of the Hill estimator and give bounds on k when the central limit theorem is still applicable. We illustrate the theory by using simulated and real-life data.     

Asymptotic distribution of a Pickands-type estimator of the extreme-value index

One of the main goals of extreme-value analysis is to estimate the probability of rare events given a sample from an unknown distribution. The upper tail behavior of this distribution is described by the extreme-value index ξ. The aim of this Note is to establish the asymptotic distribution of the estimator of $\xi \in \mathbb{R}$ introduced in Gardes and Girard [A Pickands-type estimator of the extreme-value index, Technical Report LMC-IMAG, RR-1063, 2004]. We also give its rate of convergence in some typical situations. To cite this article: L. Gardes, S. Girard, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Uniform estimator of the extremal index of stochastic recurrent sequences

A uniform upper bound of the extremal index of some stochastic recurrent sequences is obtained in the paper. We used a new approach consisting in the consideration of sequences of observations (at deterministic or random moments of time) of a continuous time process given by a stochastic differential equation.     

Estimation of the tail index in the max-aggregation scheme

In this paper, we continue the investigation of an estimator proposed in [V. Paulauskas and M. Vai?iulis, Several modifications of DPR estimator of the tail index, Lith. Math. J., 51(1):36?C50, 2011]. Specifically, we investigate the asymptotic behavior of the so-called DPR estimator under several mostly popular max-aggregation schemes.     

Consistent estimation of the tail index for dependent data

We consider the estimation of the tail-index for dependent random variables. We establish the consistency of the geometric-type estimator (Brito and Freitas, 2003) for stationary sequences satisfying general mixing conditions and derive a simplified condition, specially adapted for applications.     

On the tail index of a heavy tailed distribution

This paper proposes some new estimators for the tail index of a heavy tailed distribution when only a few largest values are observed within blocks. These estimators are proved to be asymptotically normal under suitable conditions, and their Edgeworth expansions are obtained. Empirical likelihood method is also employed to construct confidence intervals for the tail index. The comparison for the confidence intervals based on the normal approximation and the empirical likelihood method is made in terms of coverage probability and length of the confidence intervals. The simulation study shows that the empirical likelihood method outperforms the normal approximation method.     

The extent of the maximum likelihood estimator for the extreme value index

In extreme value analysis, staring from Smith (1987) [1], the maximum likelihood procedure is applied in estimating the shape parameter of tails—the extreme value index γ. For its theoretical properties, Zhou (2009) [12] proved that the maximum likelihood estimator eventually exists and is consistent for γ>−1 under the first order condition. The combination of Zhou (2009) [12] and Drees et al (2004) [11] provides the asymptotic normality under the second order condition for γ>−1/2. This paper proves the asymptotic normality for −1<γ≤−1/2 and the non-consistency for γ<−1. These results close the discussion on the theoretical properties of the maximum likelihood estimator.     

A continuous updating weighted least squares estimator of tail dependence in high dimensions

Likelihood-based procedures are a common way to estimate tail dependence parameters. They are not applicable, however, in non-differentiable models such as those arising from recent max-linear structural equation models. Moreover, they can be hard to compute in higher dimensions. An adaptive weighted least-squares procedure matching nonparametric estimates of the stable tail dependence function with the corresponding values of a parametrically specified proposal yields a novel minimum-distance estimator. The estimator is easy to calculate and applies to a wide range of sampling schemes and tail dependence models. In large samples, it is asymptotically normal with an explicit and estimable covariance matrix. The minimum distance obtained forms the basis of a goodness-of-fit statistic whose asymptotic distribution is chi-square. Extensive Monte Carlo simulations confirm the excellent finite-sample performance of the estimator and demonstrate that it is a strong competitor to currently available methods. The estimator is then applied to disentangle sources of tail dependence in European stock markets.     

Weighted least squares estimators for the Parzen tail index

Periodica Mathematica Hungarica - Estimation of the tail index of heavy-tailed distributions and its applications are essential in many research areas. We propose a class of weighted least squares...