Geodesic
Pareto approximation of the tail by local exponential modeling
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2007), pp. 3-24.

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We give a new adaptive method for selecting the number of upper order statistics used in the estimation of the tail of a distribution function. Our approach is based on approximation by an exponential model. The selection procedure consists in consecutive testing for the hypothesis of homogeneity of the estimated parameter against the change-point alternative. The selected number of upper order statistics corresponds to the first detected change-point. Our main results are non-asymptotic. 
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Ion Grama; Vladimir Spokoiny. Pareto approximation of the tail by local exponential modeling. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2007), pp. 3-24. https://geodesic-test.mathdoc.fr/item/BASM_2007_1_a0/

[1] Barron A., Birgé L., Massart P., “Risk bounds for model selection via penalization”, Probability Theory and Related Fields, 113 (1999), 301–413 | DOI | MR | Zbl

[2] Beirlant J., Teugels J., Vinysaker P., Practical analysis of extreme values, Leuven University Press, 1996 | Zbl

[3] Beirlant J., Dierskx G., Goegebeur Y., Matthys G., “Tail index estimation and an exponential regression model”, Extremes, 2:2 (1999), 177–200 | DOI | MR | Zbl

[4] Bingham N. H., Goldie C. M., Teugels J. L., Regular variation, Cambridge University Press, Cambridge, 1987 | MR | Zbl

[5] Danielsson J., de Haan L., Peng L., Vries, “Using a bootstrap method to choose the sample fraction in tail index estimation”, Journal of Multivariate Analysis, 76 (2001), 226–248 | DOI | MR | Zbl

[6] Deheuvels J., Haeusler E., Masson D., “Almost sure convergence of the Hill estimator”, Mathematical Proceedings of the Cambridge Philosophical society, 104 (1988), 371–384 | DOI | MR

[7] Drees H., “Minimax risk bounds in extreme value theory”, Ann. Statist., 29 (2001), 266–294 | DOI | MR | Zbl

[8] Drees H., Kaufman E., “Selecting the optimal sample fraction in univariate extreme value estimation”, Stoch. Processes Appl., 75 (1998), 149–172 | DOI | MR | Zbl

[9] Embrechts P., Klüppelberg K., Mikosch T., Modelling extremal events, Springer, 1997 | MR | Zbl

[10] de Haan L., Resnik S. I., “A simple asymptotic estimate for the index of a stable distribution”, J. Roy. Statist. Soc., Ser. B, 42 (1980), 83–87 | MR | Zbl

[11] Hall P., “On some simple estimates of an exponent of regular variation”, J. Roy. Statist. Soc., Ser. B, 44 (1982), 37–42 | MR | Zbl

[12] Hall P., Welsh A. H., “Adaptive estimates of regular variation”, Ann. Statist., 13 (1985), 331–341 | DOI | MR | Zbl

[13] Hill B. M., “A simple general approach to inference about the tail of a distribution”, Ann. Statist., 3 (1975), 1163–1174 | DOI | MR | Zbl

[14] Mason D., “Laws of large numbers for sums of extreme values”, Ann. Probab., 10 (1982), 754–764 | DOI | MR | Zbl

[15] Pickands J., III, “Statistical inference using extreme order statistics”, Ann. Statist., 3 (1975), 119–131 | DOI | MR | Zbl

[16] Picard D., Tribouley K., “Evolutionary Pareto distributions”, Ann. I. H. Poincaré - PR, 38 (2002), 1009–1022 | DOI | MR

[17] Reiss R.-D., Approximate distributions of order statistics: with applications to nonparametric statistics, Springer, 1989 | MR

[18] Resnik S., Starica C., “Smoothing the Hill estimator”, Adv. in Appl. Probab., 29 (1997), 271–293 | DOI | MR

[19] Resnik S., Starica C., “Tail index estimation for dependent data”, Ann. Appl. Probab., 8 (1998), 1156–1183 | DOI | MR

[20] Resnik S. I., “Heavy tail modelling and teletraffic data”, Ann. Statist., 25 (1997), 1805–1869 | DOI | MR

[21] Seneta E., Regularly varying functions, Lecture Notes in Mathematics, 508, Springer, 1976 | MR | Zbl

[22] Spokoiny V., “Estimation of a function with discontinuities via local polynomial fit with an adaptive window choice”, Ann. Statist., 26:4 (1998), 1356–1378 | DOI | MR | Zbl