Electronic Journal of Statistics

A moment estimator for the conditional extreme-value index

Gilles Stupfler

Full-text: Open access

Abstract

In extreme value theory, the so-called extreme-value index is a parameter that controls the behavior of a distribution function in its right tail. Knowing this parameter is thus essential to solve many problems related to extreme events. In this paper, the estimation of the extreme-value index is considered in the presence of a random covariate, whether the conditional distribution of the variable of interest belongs to the Fréchet, Weibull or Gumbel max-domain of attraction. The pointwise weak consistency and asymptotic normality of the proposed estimator are established. We examine the finite sample performance of our estimator in a simulation study and we illustrate its behavior on a real set of fire insurance data.

Article information

Source
Electron. J. Statist., Volume 7 (2013), 2298-2343.

Dates
First available in Project Euclid: 19 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1379596772

Digital Object Identifier
doi:10.1214/13-EJS846

Mathematical Reviews number (MathSciNet)
MR3108815

Zentralblatt MATH identifier
1293.62081

Subjects
Primary: 62G05: Estimation 62G20: Asymptotic properties 62G30: Order statistics; empirical distribution functions 62G32: Statistics of extreme values; tail inference

Keywords
Extreme-value index moment estimator random covariate consistency asymptotic normality

Citation

Stupfler, Gilles. A moment estimator for the conditional extreme-value index. Electron. J. Statist. 7 (2013), 2298--2343. doi:10.1214/13-EJS846. https://projecteuclid.org/euclid.ejs/1379596772


Export citation

References

  • [1] Beirlant, J., Boniphace, E., Dierckx, G. (2011). Generalized sum plots, REVSTAT 9(2): 181–198.
  • [2] Beirlant, J., Goegebeur, Y. (2004). Simultaneous tail index estimation, REVSTAT 2(1): 15–39.
  • [3] Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J. (2004)., Statistics of Extremes, John Wiley and Sons.
  • [4] Bingham, N.H., Goldie, C.M., Teugels, J.L. (1987)., Regular Variation, Cambridge, U.K.: Cambridge University Press.
  • [5] Chavez-Demoulin, V., Davison, A.C. (2005). Generalized additive modelling of sample extremes, Journal of the Royal Statistical Society, Series C 54: 207–222.
  • [6] Daouia, A., Florens, J.-P., Simar, L. (2010). Frontier estimation and extreme value theory, Bernoulli 16(4): 1039–1063.
  • [7] Daouia, A., Gardes, L., Girard, S., Lekina, A. (2011). Kernel estimators of extreme level curves, Test 20(2): 311–333.
  • [8] Daouia, A., Gardes, L., Girard, S. (2013). On kernel smoothing for extremal quantile regression, Bernoulli, to appear.
  • [9] Davison, A.C., Ramesh, N.I. (2000). Local likelihood smoothing of sample extremes, Journal of the Royal Statistical Society, Series B 62: 191–208.
  • [10] Davison, A.C., Smith, R.L. (1990). Models for exceedances over high thresholds, Journal of the Royal Statistical Society, Series B 52: 393–442.
  • [11] Dekkers, A.L.M., Einmahl, J.H.J., de Haan, L. (1989). A moment estimator for the index of an extreme-value distribution, Annals of Statistics 17(4): 1833–1855.
  • [12] Ferrez, J., Davison, A.C., Rebetez, M. (2011). Extreme temperature analysis under forest cover compared to an open field, Agricultural and Forest Meteorology 151: 992–1001.
  • [13] Fisher, R.A., Tippett, L.H.C. (1928). Limiting forms of the frequency distribution of the largest or smallest member of a sample, Proceedings of the Cambridge Philosophical Society 24: 180–190.
  • [14] Gardes, L., Girard, S. (2008). A moving window approach for nonparametric estimation of the conditional tail index, Journal of Multivariate Analysis 99: 2368–2388.
  • [15] Gardes, L., Girard, S. (2010). Conditional extremes from heavy-tailed distributions: An application to the estimation of extreme rainfall return levels, Extremes 13: 177–204.
  • [16] Gardes, L., Girard, S. (2012). Functional kernel estimators of large conditional quantiles, Electronic Journal of Statistics 6: 1715–1744.
  • [17] Gardes, L., Girard, S., Lekina, A. (2010). Functional nonparametric estimation of conditional extreme quantiles, Journal of Multivariate Analysis 101(2): 419–-433.
  • [18] Gardes, L., Stupfler, G. (2013). Estimation of the conditional tail-index using a smoothed local Hill estimator, Extremes, to appear.
  • [19] Gnedenko, B.V. (1943). Sur la distribution limite du terme maximum d’une série aléatoire, Annals of Mathematics 44: 423–453.
  • [20] Goegebeur, Y., Guillou, A., Schorgen, A. (2013). Nonparametric regression estimation of conditional tails – the random covariate case, Statistics, to appear.
  • [21] de Haan, L., Ferreira, A. (2006)., Extreme Value Theory: An Introduction, New York: Springer.
  • [22] Hall, P., Tajvidi, N. (2000). Nonparametric analysis of temporal trend when fitting parametric models to extreme-value data, Statistical Science 15: 153–167.
  • [23] Härdle, W., Marron, J.S. (1985). Optimal bandwidth selection in nonparametric regression function estimation, Annals of Statistics 13(4), 1465–1481.
  • [24] Hill, B.M. (1975). A simple general approach to inference about the tail of a distribution, Annals of Statistics 3: 1163–1174.
  • [25] Pickands, J. (1975). Statistical inference using extreme order statistics, Annals of Statistics 3: 119–131.
  • [26] Pisarenko, V.F., Sornette, D. (2003). Characterization of the frequency of extreme earthquake events by the generalized Pareto distribution, Pure and Applied Geophysics 160: 2343–2364.
  • [27] Smith, R.L. (1989). Extreme value analysis of environmental time series: an application to trend detection in ground-level ozone (with discussion), Statistical Science 4: 367–393.
  • [28] Wang, H., Tsai, C.L. (2009). Tail index regression, Journal of the American Statistical Association 104(487): 1233–1240.