, Volume 21, Issue 4, pp 697–729 | Cite as

Estimating an endpoint with high-order moments

  • Stéphane Girard
  • Armelle Guillou
  • Gilles Stupfler
Original Paper


We present a new method for estimating the endpoint of a unidimensional sample when the distribution function decreases at a polynomial rate to zero in the neighborhood of the endpoint. The estimator is based on the use of high-order moments of the variable of interest. It is assumed that the order of the moments goes to infinity, and we give conditions on its rate of divergence to get the asymptotic normality of the estimator. The good performance of the estimator is illustrated on some finite sample situations.


Endpoint estimation High-order moments Consistency Asymptotic normality 

Mathematics Subject Classification (2000)

62G32 62G05 



The authors are indebted to the anonymous referees for their helpful comments and suggestions that have contributed to an improved presentation of the results of this paper.


  1. Aarssen K, de Haan L (1994) On the maximal life span of humans. Math Popul Stud 4(4):259–281 zbMATHCrossRefGoogle Scholar
  2. Athreya KB, Fukuchi J (1997) Confidence intervals for endpoints of a c.d.f. via bootstrap. J Stat Plan Inference 58:299–320 MathSciNetzbMATHCrossRefGoogle Scholar
  3. Billingsley P (1979) Probability and measure. Wiley, New York zbMATHGoogle Scholar
  4. Bingham NH, Goldie CM, Teugels JL (1987) Regular variation. Cambridge University Press, Cambridge zbMATHGoogle Scholar
  5. Chow YS, Teicher H (1997) Probability theory. Springer, Berlin zbMATHCrossRefGoogle Scholar
  6. Cooke P (1979) Statistical inference for bounds of random variables. Biometrika 66:367–374 MathSciNetzbMATHCrossRefGoogle Scholar
  7. de Haan L (1981) Estimation of the minimum of a function using order statistics. J Am Stat Assoc 76:467–469 zbMATHCrossRefGoogle Scholar
  8. de Haan L, Ferreira A (2006) Extreme value theory. Springer, Berlin zbMATHGoogle Scholar
  9. Dekkers ALM, Einmahl JHJ, de Haan L (1989) A moment estimator for the index of an extreme-value distribution. Ann Stat 17:1833–1855 zbMATHCrossRefGoogle Scholar
  10. Drees H, Ferreira A, de Haan L (2003) On maximum likelihood estimation of the extreme value index. Ann Appl Probab 14:1179–1201 Google Scholar
  11. Goldenshluger A, Tsybakov A (2004) Estimating the endpoint of a distribution in the presence of additive observation errors. Stat Probab Lett 68:39–49 MathSciNetzbMATHCrossRefGoogle Scholar
  12. Hall P (1982) On estimating the endpoint of a distribution. Ann Stat 10(2):556–568 zbMATHCrossRefGoogle Scholar
  13. Hall P, Wang JZ (1999) Estimating the end-point of a probability distribution using minimum-distance methods. Bernoulli 5(1):177–189 MathSciNetzbMATHCrossRefGoogle Scholar
  14. Hall P, Wang JZ (2005) Bayesian likelihood methods for estimating the end point of a distribution. J R Stat Soc, Ser B, Stat Methodol 67(5):717–729 MathSciNetzbMATHCrossRefGoogle Scholar
  15. Hosking JRM, Wallis JR (1987) Parameter and quantile estimation for the generalized Pareto distribution. Technometrics 29:339–349 MathSciNetzbMATHCrossRefGoogle Scholar
  16. Li D, Peng L (2009) Does bias reduction with external estimator of second order parameter work for endpoint? J Stat Plan Inference 139:1937–1952 MathSciNetzbMATHCrossRefGoogle Scholar
  17. Li D, Peng L, Xu X (2011a) Bias reduction for endpoint estimation. Extremes, 14:393–412. doi: 10.1007/s10687-010-0118-2 MathSciNetCrossRefGoogle Scholar
  18. Li D, Peng L, Qi Y (2011b) Empirical likelihood confidence intervals for the endpoint of a distribution function. Test 20:353–366 MathSciNetCrossRefGoogle Scholar
  19. Loh WY (1984) Estimating an endpoint of a distribution with resampling methods. Ann Stat 12(4):1543–1550 MathSciNetzbMATHCrossRefGoogle Scholar
  20. Miller RG (1964) A trustworthy jackknife. Ann Math Stat 35:1594–1605 zbMATHCrossRefGoogle Scholar
  21. Neves C, Pereira A (2010) Detecting finiteness in the right endpoint of light-tailed distributions. Stat Probab Lett 80:437–444 MathSciNetzbMATHCrossRefGoogle Scholar
  22. Robson DS, Whitlock JH (1964) Estimation of a truncation point. Biometrika 51:33–39 MathSciNetzbMATHGoogle Scholar
  23. Smith RL (1987) Estimating tails of probability distributions. Ann Stat 15(3):1174–1207 zbMATHCrossRefGoogle Scholar
  24. Smith RL, Weissman I (1985) Maximum likelihood estimation of the lower tail of a probability distribution. J R Stat Soc, Ser B, Stat Methodol 47:285–298 MathSciNetzbMATHGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2011

Authors and Affiliations

  • Stéphane Girard
    • 1
  • Armelle Guillou
    • 2
  • Gilles Stupfler
    • 2
  1. 1.Team Mistis, INRIA Rhône-Alpes & LJK, InovalléeSaint-Ismier cedexFrance
  2. 2.Université de Strasbourg & CNRS, IRMA, UMR 7501Strasbourg cedexFrance

Personalised recommendations