Extreme Value Theory and Statistics of Univariate Extremes: A Review

type="main" xml:id="insr12058-abs-0001"> Statistical issues arising in modelling univariate extremes of a random sample have been successfully used in the most diverse fields, such as biometrics, finance, insurance and risk theory. Statistics of univariate extremes (SUE), the subject to be dealt with in this review paper, has recently faced a huge development, partially because rare events can have catastrophic consequences for human activities, through their impact on the natural and constructed environments. In the last decades, there has been a shift from the area of parametric SUE, based on probabilistic asymptotic results in extreme value theory, towards semi-parametric approaches. After a brief reference to Gumbel's block methodology and more recent improvements in the parametric framework, we present an overview of the developments on the estimation of parameters of extreme events and on the testing of extreme value conditions under a semi-parametric framework. We further discuss a few challenging topics in the area of SUE. © 2014 The Authors. International Statistical Review © 2014 International Statistical Institute

[1]  Paul Deheuvels,et al.  Kernel Estimates of the Tail Index of a Distribution , 1985 .

[2]  Janos Galambos,et al.  Rates of Convergence in Extreme Value Theory , 1984 .

[3]  Maria da Graça Temido,et al.  Rarely Observed Sample Maxima , 2001 .

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

[5]  Nader Tajvidi,et al.  Extreme value statistics and wind storm losses: a case study. , 1997 .

[6]  Philippe Naveau,et al.  IMPROVING PROBABILITY-WEIGHTED MOMENT METHODS FOR THE GENERALIZED EXTREME VALUE DISTRIBUTION , 2008 .

[7]  Luísa Canto e Castro,et al.  Generalized Pickands’ estimators for the tail index parameter and max-semistability , 2011 .

[8]  Holger Drees,et al.  Limit theorems for empirical processes of cluster functionals , 2009, 0910.0343.

[9]  L. Peng,et al.  A Bootstrap-based Method to Achieve Optimality in Estimating the Extreme-value Index , 2000 .

[10]  N. Markovich.  Nonparametric analysis of univariate heavy-tailed data , 2007 .

[11]  M. Neves,et al.  Alternatives to a Semi-Parametric Estimator of Parameters of Rare Events—The Jackknife Methodology* , 2000 .

[12]  U. Stadtmüller,et al.  Generalized regular variation of second order , 1996, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

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

[14]  Richard L. Smith.  Approximations in Extreme Value Theory. , 1987 .

[15]  Peter Hall,et al.  Using the bootstrap to estimate mean squared error and select smoothing parameter in nonparametric problems , 1990 .

[16]  S. Coles,et al.  Likelihood-Based Inference for Extreme Value Models , 1999 .

[17]  Armelle Guillou,et al.  Reduced-bias estimator of the Conditional Tail Expectation of heavy-tailed distributions , 2015 .

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

[19]  M. R. Leadbetter,et al.  Extremes and Related Properties of Random Sequences and Processes: Springer Series in Statistics , 1983 .

[20]  K. F. Turkman,et al.  A Predictive Approach to Tail Probability Estimation , 2001 .

[21]  M. Gomes,et al.  AN OVERVIEW AND OPEN RESEARCH TOPICS IN STATISTICS OF UNIVARIATE EXTREMES , 2012 .

[22]  A. McNeil.  Estimating the Tails of Loss Severity Distributions Using Extreme Value Theory , 1997, ASTIN Bulletin.

[23]  M. I. Fraga Alves.  The Influence of Central Observations on Discrimination among Multivariate Extremal Models , 1993 .

[24]  Deyuan Li,et al.  On testing extreme value conditions , 2006 .

[25]  Richard L. Smith.  Maximum likelihood estimation in a class of nonregular cases , 1985 .

[26]  Jan R. Magnus,et al.  Records in Athletics Through Extreme-Value Theory , 2006 .

[27]  Johan Segers,et al.  Second-order refined peaks-over-threshold modelling for heavy-tailed distributions , 2009, 0901.1518.

[28]  J. Johansson.  Estimating the Mean of Heavy-Tailed Distributions , 2003 .

[29]  Jonathan A. Tawn,et al.  An extreme-value theory model for dependent observations , 1988 .

[30]  Jan Beirlant,et al.  Estimating catastrophic quantile levels for heavy-tailed distributions , 2004 .

[31]  Holger Rootzén,et al.  Extremes and Related Properties of Random Sequences and Processes: Springer Series in Statistics , 1983 .

[32]  Michael A. Stephens,et al.  Tests for the Exponential Distribution , 2017 .

[33]  C. Klüppelberg,et al.  Modelling Extremal Events , 1997 .

[34]  M. Ivette Gomes,et al.  A simple generalisation of the Hill estimator , 2013, Comput. Stat. Data Anal..

[35]  Narayanaswamy Balakrishnan,et al.  Order statistics from extreme value distribution, ii: best linear unbiased estimates and some other uses , 1992 .

[36]  M.A.J. van Montfort,et al.  An asymmetric test on the type of the distribution of extremes , 1973 .

[37]  Liang Peng,et al.  Empirical likelihood confidence intervals for the endpoint of a distribution function , 2011 .

[38]  P. Embrechts,et al.  Extremes and Robustness: A Contradiction? , 2006 .

[39]  Mhamed-Ali El-Aroui,et al.  Quasi-Conjugate Bayes Estimates for GPD Parameters and Application to Heavy Tails Modelling , 2005, 1103.6216.

[40]  Carl Scarrott,et al.  A Review of Extreme Value Threshold Estimation and Uncertainty Quantification , 2012 .

[41]  Chen Zhou,et al.  The extent of the maximum likelihood estimator for the extreme value index , 2010, J. Multivar. Anal..

[42]  M. Ivette Gomes,et al.  DIRECT REDUCTION OF BIAS OF THE CLASSI- CAL HILL ESTIMATOR ⁄ , 2005 .

[43]  M. Ivette Gomes,et al.  Semi-Parametric Probability-Weighted Moments Estimation Revisited , 2012, Methodology and Computing in Applied Probability.

[44]  D. J. Goodman,et al.  Bayesian Risk Analysis , 2000 .

[45]  J. Teugels,et al.  Tail Index Estimation, Pareto Quantile Plots, and Regression Diagnostics , 1996 .

[46]  Allan J. Macleod.  A Remark on Algorithm as 215: Maximum‐Likelihood Estimation of the Parameters of the Generalized Extreme‐Value Distribution , 1989 .

[47]  Chen Zhou,et al.  Existence and consistency of the maximum likelihood estimator for the extreme value index , 2009, J. Multivar. Anal..

[48]  Jan Beirlant,et al.  Generalized Kernel Estimators for the Weibull-Tail Coefficient , 2010 .

[49]  Frederico Caeiro,et al.  Semi-parametric tail inference through probability-weighted moments , 2011 .

[50]  Frank Marohn,et al.  Testing the Gumbel Hypothesis Via the Pot-Method , 1998 .

[51]  L. Haan,et al.  On the Estimation of the Extreme-Value Index and Large Quantile Estimation , 1989 .

[52]  N. Christopeit,et al.  Estimating parameters of an extreme value distribution by the method of moments , 1994 .

[53]  A. Frigessi,et al.  A Dynamic Mixture Model for Unsupervised Tail Estimation without Threshold Selection , 2002 .

[54]  J. Hosking,et al.  Parameter and quantile estimation for the generalized pareto distribution , 1987 .

[55]  J.-P. Raoult,et al.  Rate of convergence for the generalized Pareto approximation of the excesses , 2003, Advances in Applied Probability.

[56]  J. Hüsler,et al.  Laws of Small Numbers: Extremes and Rare Events , 1994 .

[57]  M. Ivette Gomes,et al.  Penultimate Behaviour of the Extremes , 1994 .

[58]  Sándor Csörgő,et al.  SIMPLE ESTIMATORS OF THE ENDPOINT OF A DISTRIBUTION , 1989 .

[59]  J. Hüsler,et al.  Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology and Other Fields , 2007 .

[60]  J. Tiago de Oliveira,et al.  Univariate Extremes; Statistical Choice , 1984 .

[61]  J. D. T. Oliveira,et al.  The Asymptotic Theory of Extreme Order Statistics , 1979 .

[62]  M. Ivette Gomes,et al.  Statistical choice of extremal models for complete and censored data , 1985 .

[63]  J. R. Wallis,et al.  Probability Weighted Moments: Definition and Relation to Parameters of Several Distributions Expressable in Inverse Form , 1979 .

[64]  Ishay Weissman,et al.  Statistical Estimation in Extreme Value Theory , 1984 .

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

[66]  Jan Beirlant,et al.  Estimation of the extreme value index and extreme quantiles under random censoring , 2007 .

[67]  A. Otten,et al.  The power of two tests on the type of distributions of extremes , 1978 .

[68]  J. Diebolt,et al.  Approximation of the distribution of excesses through a generalized probability-weighted moments method , 2007 .

[69]  J. Stedinger,et al.  Generalized maximum‐likelihood generalized extreme‐value quantile estimators for hydrologic data , 2000 .

[70]  Frank Marohn.  An Adaptive Efficient Test for Gumbel Domain of Attraction , 1998 .

[71]  Richard L. Smith,et al.  A Comparison of Maximum Likelihood and Bayesian Estimators for the Three‐Parameter Weibull Distribution , 1987 .

[72]  J. Diebolt,et al.  Bias-reduced estimators of the Weibull tail-coefficient , 2008, 1103.6172.

[73]  P. Hall,et al.  Estimating a tail exponent by modelling departure from a Pareto distribution , 1999 .

[74]  S. Girard.  A Hill Type Estimator of the Weibull Tail-Coefficient , 2004 .

[75]  Laurens de Haan,et al.  Third order extended regular variation , 2006 .

[76]  M. Ivette Gomes.  Concomitants in a Multidimensional Extreme Model , 1984 .

[77]  M. Ivette Gomes,et al.  A new class of semi-parametric estimators of the second order parameter. , 2003 .

[78]  Johan Segers,et al.  Testing the Gumbel hypothesis by Galton's ratio , 2000 .

[79]  Alex Luiz Ferreira,et al.  Optimal asymptotic estimation of small exceedance probabilities , 2002 .

[80]  Richard L. Smith.  Estimating tails of probability distributions , 1987 .

[81]  Liang Peng,et al.  Bias reduction for high quantiles , 2010 .

[82]  Isabel Fraga Alves,et al.  ESTIMATION OF THE FINITE RIGHT ENDPOINT IN THE GUMBEL DOMAIN , 2013, 1306.1452.

[83]  M. Ivette Gomes,et al.  Subsampling techniques and the Jackknife methodology in the estimation of the extremal index , 2008, Comput. Stat. Data Anal..

[84]  J. Beirlant,et al.  A goodness-of-fit statistic for Pareto-type behaviour , 2006 .

[85]  James O. Berger,et al.  Bayesian and Frequentist Approaches to Parametric Predictive Inference , 1999 .

[86]  M. I. Gomes.  Generalized Gumbel and likelihood ratio test statistics in the multivariate GEV model , 1989 .

[87]  C. Neves,et al.  PORT-ESTIMATION OF A SHAPE SECOND-ORDER PARAMETER , 2014 .

[88]  C. Anderson.  Contributions to the asymptotic theory of extreme values , 1971 .

[89]  M. C. Jones,et al.  Robust and efficient estimation by minimising a density power divergence , 1998 .

[90]  J. Beirlant,et al.  Pareto Index Estimation Under Moderate Right Censoring , 2001 .

[91]  Fernanda Figueiredo,et al.  Bias reduction in risk modelling: Semi-parametric quantile estimation , 2006 .

[92]  Debbie J. Dupuis,et al.  A Comparison of confidence intervals for generalized extreme-value distributions , 1998 .

[93]  L. Haan,et al.  On the block maxima method in extreme value theory: PWM estimators , 2013, 1310.3222.

[94]  M. Ivette Gomes,et al.  Approximation by Penultimate Extreme Value Distributions , 1998 .

[95]  A. M. Hasofer,et al.  A Test for Extreme Value Domain of Attraction , 1992 .

[96]  M. Ivette Gomes,et al.  Statistical choice of extreme value domains of attraction — a comparative analysis , 1996 .

[97]  Sander Smeets,et al.  Ultimate 100‐m world records through extreme‐value theory , 2009 .

[98]  John H. J. Einmahl,et al.  Ultimate 100-m world records through extreme-value theory , 2011 .

[99]  M. Ivette Gomes,et al.  Reduced-Bias Location-Invariant Extreme Value Index Estimation: A Simulation Study , 2011, Commun. Stat. Simul. Comput..

[100]  A. Walden,et al.  Maximum likelihood estimation of the parameters of the generalized extreme-value distribution , 1980 .

[101]  R. Rackwitz,et al.  On predictive distribution functions for the three asymptotic extreme value distributions , 1992 .

[102]  M. Ivette Gomes,et al.  Adaptive PORT–MVRB estimation: an empirical comparison of two heuristic algorithms , 2013 .

[103]  Jean-Noël Bacro,et al.  A statistical test procedure for the shape parameter of a generalized Pareto distribution , 2004, Comput. Stat. Data Anal..

[104]  A. Jenkinson.  The frequency distribution of the annual maximum (or minimum) values of meteorological elements , 1955 .

[105]  Leal De Carvalho Gomes,et al.  Some probabilistic and statistical problems in extreme value theory , 1978 .

[106]  M. Ivette Gomes,et al.  An I-Dimensional Limiting Distribution Function of Largest Values and Its Relevance to the Statistical Theory of Extremes , 1981 .

[107]  Vytaras Brazauskas,et al.  Robust Estimation of Tail Parameters for Two-Parameter Pareto and Exponential Models via Generalized Quantile Statistics , 2000 .

[108]  R. Fisher,et al.  Limiting forms of the frequency distribution of the largest or smallest member of a sample , 1928, Mathematical Proceedings of the Cambridge Philosophical Society.

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

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

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

[112]  Armelle Guillou,et al.  An asymptotically unbiased minimum density power divergence estimator for the Pareto-tail index , 2013, J. Multivar. Anal..

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

[114]  Stuart G. Coles,et al.  Bayesian methods in extreme value modelling: a review and new developments. , 1996 .

[115]  A. Hadi,et al.  Fitting the Generalized Pareto Distribution to Data , 1997 .

[116]  Jan Beirlant,et al.  Excess functions and estimation of the extreme-value index , 1996 .

[117]  Jonathan A. Tawn,et al.  A Bayesian Analysis of Extreme Rainfall Data , 1996 .

[118]  L. Haan,et al.  Bias correction in extreme value statistics with index around zero , 2013 .

[119]  Frederico Caeiro,et al.  An Overview And Open Research Topics In Statistics Of Univariate Extremes , 2012 .

[120]  Peter Hall,et al.  On Estimating the Endpoint of a Distribution , 1982 .

[121]  M. Fréchet.  Sur la loi de probabilité de l'écart maximum , 1928 .

[122]  Liang Peng,et al.  Semi-parametric Estimation of the Second Order Parameter in Statistics of Extremes , 2002 .

[123]  Michael A. Stephens,et al.  Goodness of fit for the extreme value distribution , 1977 .

[124]  M. Ivette Gomes,et al.  Reduced-Bias Tail Index Estimators Under a Third-Order Framework , 2009 .

[125]  G. S. Lingappaiah.  Bayesian Prediction Regions for the Extreme Order Statistics , 1984 .

[126]  Björn Vandewalle,et al.  A heuristic adaptive choice of the threshold for bias-corrected Hill estimators , 2008 .

[127]  Liang Peng,et al.  Asymptotically unbiased estimators for the extreme-value index , 1998 .

[128]  Robert Kinnison,et al.  Correlation Coefficient Goodness-of-Fit Test for the Extreme-Value Distribution , 1989 .

[129]  Michael Falk,et al.  A LAN based Neyman smooth test for Pareto distributions , 2008 .

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

[131]  M. F. Brilhante.  EXPONENTIALITY VERSUS GENERALIZED PARETO — A RESISTANT AND ROBUST TEST , 2004 .

[132]  S. Resnick.  Extreme Values, Regular Variation, and Point Processes , 1987 .

[133]  M. Ivette Gomes,et al.  HIGH QUANTILE ESTIMATION AND THE PORT METHODOLOGY , 2009 .

[134]  D. Walshaw.  Modelling extreme wind speeds in regions prone to hurricanes , 2000 .

[135]  Liang Peng,et al.  Does bias reduction with external estimator of second order parameter work for endpoint , 2009 .

[136]  J. A. Achcar,et al.  Transformation of Survival Data to an Extreme Value Distribution , 1987 .

[137]  J. Hosking.  Maximum‐Likelihood Estimation of the Parameters of the Generalized Extreme‐Value Distribution , 1985 .

[138]  J. Teugels,et al.  Practical Analysis of Extreme Values , 1996 .

[139]  Richard L. Smith.  Extreme value theory based on the r largest annual events , 1986 .

[140]  Jan Beirlant,et al.  On Exponential Representations of Log-Spacings of Extreme Order Statistics , 2002 .

[141]  James Pickands,et al.  Bayes Quantile Estimation and Threshold Selection for the Generalized Pareto Family , 1994 .

[142]  Lei Si Ni Ke Resnick.S.I..  Extreme values. regular variation. and point processes , 2011 .

[143]  Laurens de Haan,et al.  Approximations to the tail empirical distribution function with application to testing extreme value conditions , 2006 .

[144]  J. Doob.  Stochastic processes , 1953 .

[145]  Vijay P. Singh,et al.  Parameter estimation for 2-parameter generalized pareto distribution by POME , 1997 .

[146]  L. Canto e Castro,et al.  MAX-SEMISTABLE LAWS IN EXTREMES OF STATIONARY RANDOM SEQUENCES ∗ , 2003 .

[147]  M. I. Fraga Alves,et al.  A Location Invariant Hill-Type Estimator , 2001 .

[148]  Enrique Castillo,et al.  The Selection of the Domain of Attraction of an Extreme Value Distribution from a Set of Data , 1989 .

[149]  M. Ivette Gomes,et al.  Mixed moment estimator and location invariant alternatives , 2009 .

[150]  M. Ivette Gomes,et al.  Penultimate Approximations in Statistics of Extremes and Reliability of Large Coherent Systems , 2015 .

[151]  H. N. Nagaraja,et al.  Order Statistics, Third Edition , 2005, Wiley Series in Probability and Statistics.

[152]  Hogeschool-Universiteit Brussel,et al.  GENERALIZED SUM PLOTS , 2011 .

[153]  A. O'Hagan,et al.  Accounting for threshold uncertainty in extreme value estimation , 2006 .

[154]  E. Castillo.  Extreme value and related models with applications in engineering and science , 2005 .

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

[156]  Anthony C. Davison,et al.  Modelling Excesses over High Thresholds, with an Application , 1984 .

[157]  L. Peng.  Estimating the mean of a heavy tailed distribution , 2001 .

[158]  Laurens de Haan,et al.  On maximum likelihood estimation of the extreme value index , 2004, math/0407062.

[159]  Julian Z. Wang.  Selection of the k Largest Order Statistics for the Domain of Attraction of the Gumbel Distribution , 1995 .

[160]  Petros Dellaportas,et al.  Bayesian Analysis of Extreme Values by Mixture Modeling , 2003 .

[161]  J. Tiago de Oliveira,et al.  Statistical Choice of Univariate Extreme Models , 1981 .

[162]  M. Ivette Gomes,et al.  Concomitants and linear estimators in an i-dimensional extremal model , 1985 .

[163]  M. R. Leadbetter,et al.  On Exceedance Point Processes for Stationary Sequences under Mild Oscillation Restrictions , 1989 .

[164]  J. Corcoran.  Modelling Extremal Events for Insurance and Finance , 2002 .

[165]  Sidney I. Resnick,et al.  How to make a Hill Plot , 2000 .

[166]  Jan Beirlant,et al.  Estimation of the extreme-value index and generalized quantile plots , 2005 .

[167]  S. Grimshaw.  Computing Maximum Likelihood Estimates for the Generalized Pareto Distribution , 1993 .

[168]  M. Ivette Gomes,et al.  A note on statistical choice of extremal models , 1982 .

[169]  Michael Falk,et al.  On testing the extreme value index via the pot-method , 1995 .

[170]  M. I. Fraga Alves.  A Location Invariant Hill-Type Estimator , 2001 .

[171]  Edgar Kaufmann.  Penultimate Approximations in Extreme Value Theory , 2000 .

[172]  Rolf-Dieter Reiss,et al.  A New Class of Bayesian Estimators in Paretian Excess-of-Loss Reinsurance , 1999, ASTIN Bulletin.

[173]  J. Hüsler,et al.  Minimum distance estimators in extreme value distributions , 1996 .

[174]  Alberto Luceño,et al.  Fitting the generalized Pareto distribution to data using maximum goodness-of-fit estimators , 2006, Comput. Stat. Data Anal..

[175]  L. de Haan,et al.  On the estimation of the exceedance probability of a high level , 1990 .

[176]  Vartan Choulakian,et al.  Goodness-of-Fit Tests for the Generalized Pareto Distribution , 2001, Technometrics.

[177]  Jan Picek,et al.  The contribution of the maximum to the sum of excesses for testing max-domains of attraction , 2006 .

[178]  M. A. Amaral Turkman,et al.  Bayesian approach to parameter estimation of the generalized pareto distribution , 2003 .

[179]  Jan Beirlant,et al.  Estimation of the Extreme Value Index , 2016 .

[180]  P. Prescott,et al.  Maximum likeiihood estimation of the parameters of the three-parameter generalized extreme-value distribution from censored samples , 1983 .

[181]  Jonathan A. Tawn,et al.  Bayesian Inference for Extremes: Accounting for the Three Extremal Types , 2004 .

[182]  J. R. Wallis,et al.  Estimation of the generalized extreme-value distribution by the method of probability-weighted moments , 1985 .

[183]  J. Z. Wang,et al.  DETERMINATION OF DOMAINS OF ATTRACTION BASED ON A SEQUENCE OF MAXIMA , 1996 .

[184]  J. Hosking.  Testing whether the shape parameter is zero in the generalized extreme-value distribution , 1984 .

[185]  Arnoldo Frigessi,et al.  Practical Extreme Value Modelling of Hydrological Floods and Droughts: A Case Study , 2004 .

[186]  George Michailidis,et al.  A Diagnostic Plot for Estimating the Tail Index of a Distribution , 2004 .

[187]  Jana Jurečková,et al.  A Class of Tests on the Tail Index , 2001 .

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

[189]  Laurens de Haan,et al.  On the estimation of high quantiles , 1993 .

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

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

[192]  E. Gumbel,et al.  Statistics of extremes , 1960 .

[193]  Cláudia Neves.  TESTING EXTREME VALUE CONDITIONS — AN OVERVIEW AND RECENT APPROACHES , 2008 .

[194]  M. I. Gomes,et al.  Exponentiality versus generalized Pareto, quick tests. , 1986 .

[195]  Frank Marohn,et al.  Testing Extreme Value Models , 2000 .

[196]  J. Geluk.  Π-regular variation , 1981 .

[197]  Laurent Gardes,et al.  Bias-reduced extreme quantile estimators of Weibull tail-distributions , 2008, 1103.6204.

[198]  Jan Beirlant,et al.  Tail Index Estimation and an Exponential Regression Model , 1999 .

[199]  Piet Groeneboom,et al.  Kernel-type estimators for the extreme value index , 2003 .

[200]  Fernanda Figueiredo,et al.  Adaptive estimation of heavy right tails: resampling-based methods in action , 2012 .

[201]  J. R. Wallis,et al.  Probability weighted moments compared with some traditional techniques in estimating Gumbel Parameters and quantiles , 1979 .

[202]  Jan Beran,et al.  The harmonic moment tail index estimator: asymptotic distribution and robustness , 2014 .

[203]  Víctor Leiva,et al.  On an extreme value version of the Birnbaum-Saunders distribution , 2012 .

[204]  Richard L. Smith.  Threshold Methods for Sample Extremes , 1984 .

[205]  M. Crovella,et al.  Estimating the Heavy Tail Index from Scaling Properties , 1999 .

[206]  Jan Beirlant,et al.  Peaks-Over-Threshold Modeling Under Random Censoring , 2010 .

[207]  Liang Peng,et al.  Robust Estimation of the Generalized Pareto Distribution , 2001 .

[208]  Jan Beirlant,et al.  ESTIMATING THE EXTREME VALUE INDEX AND HIGH QUANTILES WITH EXPONENTIAL REGRESSION MODELS , 2003 .

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

[210]  Armelle Guillou,et al.  ASYMPTOTIC BEHAVIOUR OF REGULAR ESTIMATORS , 2022 .

[211]  J. Teugels,et al.  Statistics of Extremes , 2004 .

[212]  J. V. Witter,et al.  Testing exponentiality against generalised Pareto distribution , 1985 .

[213]  Laurens de Haan,et al.  Slow Variation and Characterization of Domains of Attraction , 1984 .

[214]  M. Ivette Gomes,et al.  Two Test Statistics for Choice of Univariate Extreme Models , 1984 .

[215]  R. Reiss,et al.  Statistical Analysis of Extreme Values-with applications to insurance , 1997 .

[216]  I. V. Grinevigh.  Domains of Attraction of the Max-Semistable Laws under Linear and Power Normalizations , 1994 .

[217]  William R. Schucany,et al.  Robust and Efficient Estimation for the Generalized Pareto Distribution , 2004 .

[218]  W K Fung,et al.  Method of medians for lifetime data with Weibull models. , 1999, Statistics in medicine.

[219]  Armelle Guillou,et al.  A diagnostic for selecting the threshold in extreme value analysis , 2001 .

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

[221]  L. de Haan,et al.  On the maximal life span of humans. , 1994, Mathematical population studies.

[222]  Armelle Guillou,et al.  Statistics of Extremes Under Random Censoring , 2006, 0803.2162.

[223]  M. Gomes,et al.  Statistics of extremes for IID data and breakthroughs in the estimation of the extreme value index: Laurens de Haan leading contributions , 2008 .

[224]  Herbert A. David,et al.  Order Statistics, Third Edition , 2003, Wiley Series in Probability and Statistics.

[225]  B. Arnold,et al.  A first course in order statistics , 1994 .

[226]  S. Resnick.  Heavy tail modeling and teletraffic data: special invited paper , 1997 .

[227]  L. Haan,et al.  Residual Life Time at Great Age , 1974 .

[228]  John Lamperti,et al.  On Extreme Order Statistics , 1964 .

[229]  Andreas Christmann,et al.  A robust estimator for the tail index of Pareto-type distributions , 2007, Comput. Stat. Data Anal..

[230]  Cláudia Neves,et al.  Semi-parametric approach to the Hasofer–Wang and Greenwood statistics in extremes , 2007 .

[231]  Maria da Graça Temido,et al.  Looking for max-semistability: A new test for the extreme value condition , 2011 .

[232]  M. Ivette Gomes,et al.  The Bootstrap Methodology in Statistics of Extremes—Choice of the Optimal Sample Fraction , 2001 .

[233]  M.A.J. Van Montfort,et al.  On testing that the distribution of extremes is of type I when type II is the alternative , 1970 .

[234]  D. Farnsworth.  A First Course in Order Statistics , 1993 .

[235]  Michael A. Stephens,et al.  Asymptotic Results for Goodness-of-Fit Statistics with Unknown Parameters , 1976 .

[236]  M. Ivette Gomes,et al.  Inference in a Multivariate Generalized Extreme Value Model-Asymptotic Properties of Two Test Statistics , 1986 .

[237]  E. Pancheva,et al.  Max-semistability: a survey , 2010 .

[238]  Liang Peng,et al.  Review of testing issues in extremes: in honor of Professor Laurens de Haan , 2008 .

[239]  Saralees Nadarajah,et al.  Location invariant Weiss-Hill estimator , 2012 .

[240]  W. E. Bardsley.  A test for distinguishing between extreme value distributions , 1977 .

[241]  Cl'ement Dombry,et al.  Maximum likelihood estimators for the extreme value index based on the block maxima method , 2013, 1301.5611.

[242]  M. I. Fraga Alves.  Asymptotic distribution of Gumbel statistic in a semi-parametric approach. , 1999 .

[243]  J. Pickands.  Statistical Inference Using Extreme Order Statistics , 1975 .

[244]  Eike Christian Brechmann,et al.  Bayesian Risk Analysis , 2014 .

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

[246]  L. Haan,et al.  Using a Bootstrap Method to Choose the Sample Fraction in Tail Index Estimation , 2000 .

[247]  D. Dupuis.  Exceedances over High Thresholds: A Guide to Threshold Selection , 1999 .

[248]  E. J. Gumbel,et al.  A Quick Estimation of the Parameters in Frechet's Distribution , 1965 .

[249]  Richard L. Smith,et al.  Models for exceedances over high thresholds , 1990 .

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

[251]  Edgar Kaufmann,et al.  Selecting the optimal sample fraction in univariate extreme value estimation , 1998 .

[252]  M.A.J. van Montfort,et al.  On testing a shape parameter in the presence of a location and a scale parameter , 1978 .

[253]  Michael Falk,et al.  LAN of extreme order statistics , 1995 .

[254]  Frank Marohn,et al.  ON TESTING THE EXPONENTIAL AND GUMBEL DISTRIBUTION , 1994 .

[255]  L. Haan.  On regular variation and its application to the weak convergence of sample extremes , 1973 .

[256]  K. P. Hapuarachchi,et al.  Bayes estimation of the extreme-value reliability function , 1993 .

[257]  M. Ivette Gomes,et al.  Comparison of Extremal Models through Statistical Choice in Multidimensional Backgrounds , 1989 .