HIGH QUANTILE ESTIMATION AND THE PORT METHODOLOGY

• In many areas of application, a typical requirement is to estimate a high quantile �1 p of probability 1 −p, a value, high enough, so that the chance of an exceedance of that value is equal to p, small. The semi-parametric estimation of high quantiles depends not only on the estimation of the tail index , the primary parameter of extreme events, but also on an adequate estimation of a scale first order parameter. The great majority of semi-parametric quantile estimators, in the literature, do not enjoy the adequate behaviour, in the sense that they do not suffer the appropriate linear shift in the presence of linear transformations of the data. Recently, and for heavy tails ( > 0), a new class of quantile estimators was introduced with such a behaviour. They were named PORT-quantile estimators, with PORT standing for peaks over random threshold. In this paper, also for heavy tails, we introduce a new class of PORT-quantile estimators with the above mentioned behaviour, using the PORT methodology and incorporating Hill and moment PORT-classes of tail index estimators in one of the most recent classes of quantile estimators suggested in the literature. Under convenient restrictions on the underlying model, these classes of estimators are consistent and asymptotically normal for adequate k, the number of top order statistics used in the semi-parametric estimation of �1 p.

[1]  Alan H. Welsh,et al.  Adaptive Estimates of Parameters of Regular Variation , 1985 .

[2]  Natalia M. Markovich.  Nonparametric Analysis of Univariate Heavy-Tailed Data: Research and Practice , 2007 .

[3]  Liang Peng,et al.  Comparison of tail index estimators , 1998 .

[4]  L. Haan,et al.  A moment estimator for the index of an extreme-value distribution , 1989 .

[5]  S. Resnick.  Heavy-Tail Phenomena: Probabilistic and Statistical Modeling , 2006 .

[6]  M. Ivette Gomes,et al.  Semi-parametric second-order reduced-bias high quantile estimation , 2009 .

[7]  M. Ivette Gomes,et al.  A Sturdy Reduced-Bias Extreme Quantile (VaR) Estimator , 2007 .

[8]  L. Haan,et al.  Extreme value theory : an introduction , 2006 .

[9]  Laurens de Haan,et al.  On regular variation and its application to the weak convergence of sample extremes , 1973 .

[10]  P. Hall.  On Some Simple Estimates of an Exponent of Regular Variation , 1982 .

[11]  M. Ivette Gomes,et al.  Peaks over random threshold methodology for tail index and high quantile estimation , 2006 .

[12]  A. Dekkers,et al.  Optimal choice of sample fraction in extreme-value estimation , 1993 .

[13]  I. Weissman.  Estimation of Parameters and Large Quantiles Based on the k Largest Observations , 1978 .

[14]  B. Gnedenko.  Sur La Distribution Limite Du Terme Maximum D'Une Serie Aleatoire , 1943 .

[15]  M. Gomes,et al.  Revisiting the Role of the Jackknife Methodology in the Estimation of a Positive Tail Index , 2005 .

[16]  M. Ivette Gomes,et al.  Tail index estimation for heavy‐tailed models: accommodation of bias in weighted log‐excesses , 2007 .

[17]  L. Haan,et al.  On optimising the estimation of high quantiles of a probability distribution , 2003 .

[18]  M. Ivette Gomes,et al.  IMPROVING SECOND ORDER REDUCED BIAS EXTREME VALUE INDEX ESTIMATION , 2007 .

[19]  On the extremes of randomly sub-sampled time series , 2008 .

[20]  M. Gomes,et al.  Generalizations of the Hill estimator – asymptotic versus finite sample behaviour☆ , 2001 .

[21]  B. M. Hill,et al.  A Simple General Approach to Inference About the Tail of a Distribution , 1975 .

[22]  Fernanda Figueiredo,et al.  Improved reduced-bias tail index and quantile estimators , 2008 .

[23]  M. Gomes,et al.  Asymptotic comparison of the mixed moment and classical extreme value index estimators , 2008 .

[24]  J. Geluk,et al.  Regular variation, extensions and Tauberian theorems , 1987 .

[25]  M. Gomes,et al.  Bias Reduction in Risk Modelling: Semi-Parametric , 2006 .

[26]  M. Gomes,et al.  Reduced‐bias tail index estimation and the jackknife methodology , 2007 .

[27]  S. Berman.  On Regular Variation and Its Application to the Weak Convergence of Sample Extremes , 1972 .

[28]  M. Ivette Gomes,et al.  PORT Hill and Moment Estimators for Heavy-Tailed Models , 2008, Commun. Stat. Simul. Comput..