A class of asymptotically unbiased semi-parametric estimators of the tail index

Communications in Statistics - Theory and Methods

Pereira

Miranda

2005

In this article, and in a context of regularly varying tails, we analyze a generalization of the classical Hill estimator of a positive tail index. The members of this general class of estimators are not asymptotically more eficient than the original one, the Hill estimator. We thus propose a class of generalized Jackknife estimators associated with any two members of the first class. They enable the reduction of the main component of bias of the Hill estimator and are dependent of a tuning parameter, which is adequately chosen through an asymptotic variance minimization criterion. These new estimators are compared with the Hill estimator, both asymptotically and for finite samples and, when the underlying distribution is in Hall's class of models, they really improve on the well-known, bias-variance, trade-off characteristic of the Hill estimator.

Section: Generalized Jackknife Estimators Of the Tail Indexmentioning

confidence: 94%

Section: Moreover Under the Second-order Framework In (15) We Have mentioning

confidence: 99%

Revisiting the Role of the Jackknife Methodology in the Estimation of a Positive Tail Index

Communications in Statistics - Theory and Methods

Pereira

Miranda

2005

“…This has recently led researchers to consider the possibility of dealing with the bias term in an appropriate way, building new estimators, γ R (k) say, the so-called second-order reduced-bias estimators discussed by Peng (1998), Beirlant, Dierckx, Goegebeur, and Matthys (1999), Feuerverger and Hall (1999), Gomes, Martins, and Neves (2000), Caeiro and Gomes (2002), Gomes, Figueiredo, and Mendonça (2004c), among others. Then, if (1.3) holds for models in (1.6) and for the tail index estimators introduced in the previously mentioned articles, we may write, with P R k an asymptotically standard normal random variable (rv),…”

Section: Second-order Reduced-bias Tail Index Estimationmentioning

confidence: 98%

A Sturdy Reduced-Bias Extreme Quantile (VaR) Estimator

Journal of the American Statistical Association

Pestana

2007

113

106

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 (VaR p ), 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 VaR p 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.

“…Better not to work only with one estimator; draw sample paths associated to a set of different semi-parametric estimators (i.e., functions of k, the number of order statistics involved in the estimation procedures); Estimators which are explicitly expressed in the observations are much easier to obtain and may have high efficiency provided we proceed to some kind of bias reduction, like the ones suggested by Drees (1996), Peng (1998), Beirlant et al (1999), Feuerverger and Hall (1999), Gomes et al (2000, , Caeiro and Gomes (2001), among others.…”

Section: A Semi-parametric Context 297mentioning

confidence: 98%

Maximum likelihood revisited under a semi-parametric context - estimation of the tail index

Journal of Statistical Computation and Simulation

Oliveira

2003

In this paper, and in a context of regularly varying tails, we study computationally the classical Maximum Likelihood (ML) estimator based on the Paretian behaviour of the excesses over a high threshold, denoted PML-estimator, a type II Censoring estimator based specifically on a Fréchet parent, denoted CENS-estimator, and two ML estimators based on the scaled log-spacings, and denoted SLS-estimators. These estimators are considered under a semi-parametric set-up, and compared with the classical Hill estimator and a Generalized Jackknife (GJ) estimator, which has essentially in mind a reduction of the bias of Hill's estimator.