Methodology and Computing in Applied Probability

, Volume 8, Issue 3, pp 321–343 | Cite as

A Nonparametric Sequential Test with Power 1 for the Mean of Lévy-stable Laws with Infinite Variance

  • Abdelhakim Necir


A nonparametric sequential test with power one for the mean of Lévy-stable laws with infinite variance is given. Our considerations are based on a law of the iterated logarithm for Peng’s estimator [Peng, Stat. Probab. Lett., 52:255–264, 2001] of the mean of heavy-tailed distributions. Our main motivation comes from applications to financial data, and in particular to sequential control of daily asset returns.


Heavy tails Hill’s estimator Lévy-stable law Law of the iterated logarithm Sequential test Asset returns 

AMS 2000 Subject Classification

Primary 62G32 Secondary 62G05, 62G20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. P. Carr, H. Geman, D. B. Madan, and M. Yor, “The fine structure of asset returns: an empirical investigation,” Journal of Business vol. 75 pp. 305–332, 2002.CrossRefGoogle Scholar
  2. P. L. Conti, “A nonparametric sequential test with power 1 for the ruin probability in some risk models,” Statistics & Probability Letters vol. 72 pp. 333–343, 2005.MATHMathSciNetCrossRefGoogle Scholar
  3. S. Csörgő, L. Horvàth, and D. Mason, “What portion of the sample partial sum asymptotically stable or normal?,” Probability Theory and Related Fields vol. 72 pp. 1–16, 1986.MathSciNetCrossRefGoogle Scholar
  4. L. de Haan, and U. Stadtmüler, “Generalized regular variation of second order,” Journal of the Australian Mathematical Society vol. 61 pp. 381–395, 1996.MATHGoogle Scholar
  5. L. de Hann, L. Peng, and C. G. de Vries, “Using a bootstrap method to choose the sample fraction in tail estimation,” Journal of Multivariate Analysis vol. 76 pp. 226–248, 2001.MathSciNetCrossRefGoogle Scholar
  6. P. Deheuvels, E. Haeusler, and D. Mason, “Almost sure convergence of the Hill estimator,” Mathematical Proceedings of the Cambridge Philosophical Society vol. 104 pp. 371–381, 1988.MATHMathSciNetGoogle Scholar
  7. J. H. Einmahl, and D. Mason, “Strong limit theorems for weighted quantile process,” Annals of Probability vol. 16 pp. 1623–1643, 1988.MATHMathSciNetGoogle Scholar
  8. E. F. Fama, “The behavior of stock market prices,” Journal of Business vol. 38 pp. 34–105, 1965.CrossRefGoogle Scholar
  9. E. Haeusler, and D. Mason, “Laws of the iterated logarithm for sums of the middle portion of the sample,” Mathematical Proceedings of the Cambridge Philosophical Society vol. 101 p. 301, 1987.MathSciNetCrossRefGoogle Scholar
  10. B. M. Hill, “A simple approach to inference about the tail of a distribution,” Annals of Statistics vol. 3 pp. 1163–1174, 1975.MATHMathSciNetGoogle Scholar
  11. J. Kiefer, “Iterated logarithm analogues for sample quantiles when p n↓0. In: Proceedings of the Sixth Berkeley Symposium,” Mathematical Statistics and Probability vol. 1 pp. 227–244, University of California Press, Berkeley, California, 1972.Google Scholar
  12. P. Lévy, Calcul des Probabilités, Gauthier Villars: Paris, France 1925.MATHGoogle Scholar
  13. B. B. Mandelbrot, “The variation of certain speculative prices,” Journal of Business vol. 36 pp. 394–419, 1963.CrossRefGoogle Scholar
  14. A. Necir, “A functional law of the iterated logarithm for kernel type estimators of the tail index,” Journal of Statistical Planning and Inference vol. 136 pp. 780-802, 2006.MATHMathSciNetCrossRefGoogle Scholar
  15. L. Peng, “Estimating the mean of a heavy-tailed distribution,” Statistics & Probability Letters vol. 52 pp. 255–264, 2001.MATHMathSciNetCrossRefGoogle Scholar
  16. H. Robbins, “Statistical methods related to the law of the iterated logarithm,” Annals of Mathematical Statistics vol. 41 pp. 1397–1410, 1970.MATHMathSciNetGoogle Scholar
  17. H. Robbins, and D. Siegmund, “Bound crossing probabilities for the Wiener processes and sample sums,” Annals of Mathematical Statistics vol. 41 pp. 1410–1429, 1970.MATHMathSciNetGoogle Scholar
  18. H. Robbins, and D. Siegmund, “The expected sample size of some tests with power one,” Annals of Statistics vol. 2 pp. 515–536, 1974.MathSciNetGoogle Scholar
  19. G. Samorodnitsky, and M. S. Taqqu, Stable Non-Gaussian Random Processes. Chapman & Hall: New York, 1994.MATHGoogle Scholar
  20. P. K. Sen, Sequential Nonparametrics. Wiley: New York, 1981.MATHGoogle Scholar
  21. R. Weron, “Lévy-stable distributions revisited: tail index > 2 does not exclude the Levy-stable regime,” International Journal of Modern Physics vol. C12 pp. 209–223, 2001.Google Scholar

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.Laboratory of Applied MathematicsBiskraAlgeria

Personalised recommendations