Abstract
A new class of semiparametric estimators of the tail index is proposed. These estimators are based on a rather general class of semiparametric statistics. Their asymptotic normality is proved. The new estimators are compared with several other recently introduced estimators of the tail index in terms of the asymptotic mean-square error. An algorithm to calculate the new estimators is developed and then applied to several real data sets.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Gomes, M.I. and Guillou, A., Extreme Value Theory and Statistics of Univariate Extremes: A Review. Int. Stat. Rev., 2015, no. 83, pp. 263–292.
Paulauskas, V. and Vaiciulis, M., Several New Tail Index Estimators, Ann. Inst. Stat. Math., 2017, no. 69, pp. 461–487.
Hill, B.M., A Simple General Approach to Inference about the Tail of a Distribution, Ann. Stat., 1975, no. 3, pp. 1163–1174.
Danielsson, J., Jansen, D.W., and de Vries, C.G., The Method of Moments Ratio Estimator for the Tail Shape Parameter, Commun. Stat. Theory, 1986, no. 25, pp. 711–720.
Segers, J., Residual Estimators, J. Stat. Plan. Inf., 2001, no. 98, pp. 15–27.
De Haan, L. and Peng, L., Comparison of Tail Index Estimators, Stat. Nederl, 1998, no. 52, pp. 60–70.
De Haan, L. and Ferreira, A., Extreme Value Theory: An Introduction, New York: Springer, 2006.
Paulauskas, V. and Vaiciulis, M., On the Improvement of Hill and Some Other Estimators, Lith. Math. J., 2013, no. 53, pp. 336–355.
Brilhante, F., Gomes, M.I., and Pestana, D., A Simple Generalization of the Hill Estimator, Comput. Stat. Data Anal., 2013, no. 57, pp. 518–535.
Beran, J., Schell, D., and Stehlik, M., The Harmonic Moment Tail Index Estimator: Asymptotic Distribution and Robustness, Ann. Inst. Stat. Math., 2014, no. 66, pp. 193–220.
Gomes, M.I. and Martins, M.J., Eficient Alternatives to the Hill Estimator, Proc. Workshop V.E.L.A. Extreme Values and Additive Laws. C.E.A.U.L. Ed., 1999, no. 9, pp. 40–43.
Gomes, M.I., Martins, M.J., and Neves, M., Alternatives to a Semi-parametric Estimator of Parameters of Rare Events—the Jackknife Methodology, Extremes, 2000, no. 3, pp. 207–229.
Hall, P. and Welsh, A.H., Adaptive Estimates of Parameters of Regular Variation, Ann. Statist., 1985, no. 13, pp. 331–341.
Hall, P., On Some Simple Estimates of an Exponent of Regular Variation, J. Royal Statist. Soc. B, 1982, no. 44, pp. 37–42.
Fraga Alves, M.I., Gomes, M.I., and de Haan, L., A New Class of Semi-parametric Estimators of the Second Order Parameter, Portugaliae Math., 2003, no. 60, pp. 193–214.
Gomes, M.I. and Martins, M.J., Asymptotically Unbiased Estimators of the Tail Index Based on External Estimation of the Second Order Parameter, Extremes, 2002, no. 5, pp. 5–31.
Caeiro, F. and Gomes, M.I., Minimum-variance Reduced-bias Tail Index and High Quantile Estimation, Revstat., 2008, no. 6, pp. 1–20.
Das, B. and Resnick, S., QQ Plots, Random Sets and Data from a Heavy Tailed Distribution, Stochast. Models, 2008, no. 24, pp. 103–132.
Paxson, V. and Floyd, S., Wide-area Traffic: The Failure of Poisson Modeling, IEEE/ACM Trans. Networking, 1995, no. 3, pp. 226–244.
Guo, L., Crovella, M., and Matta, I., TCP Congestion Control and Heavy Tails, Technical Report BUCS-2000-017, Computer Science Department, Boston University, 2000.
Volkovich, Y.V. and Litvak, N., Asymptotic Analysis for Personalized Web Search, Adv. Appl. Prob., 2010, no. 42 (2), pp. 577–604.
Leskovec, J. and Krevl, A., SNAP Datasets: Stanford Large Network Dataset Collection, 2014. http://snap.stanford.edu/data
Mikosch, TV., Modeling Dependence and Tails of Financial Time Series, K0benhavns Univ.: H.C.0.-Tryk, 2002, pp. 1–75.
Galbraith, J.W., Circuit Breakers and the Tail Index of Equity Returns, J. Financ. Economet., 2004. no. 2(1), pp. 109–129.
Resnick, S.I., Heavy-Tail Phenomena. Probabilistic and Statistical Modeling, New York: Springer, 2006.
Whitt, W., Stochastic-Process Limits. An Introduction to Stochastic-Process Limits and Their Application to Queues, New York: Springer, 2002.
Mitrinović, D.S., Elementary Inequalities, Groningen: P. Noordhoff, 1964.
Author information
Authors and Affiliations
Corresponding author
Additional information
Russian Text © The Author(s), 2019, published in Avtomatika i Telemekhanika, 2019, No. 10, pp. 62–77.
Rights and permissions
About this article
Cite this article
Vaičiulis, M., Markovich, N.M. A Class of Semiparametric Tail Index Estimators and Its Applications. Autom Remote Control 80, 1803–1816 (2019). https://doi.org/10.1134/S0005117919100035
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117919100035