Abstract
In this note, we prove a characterization of extreme value distributions. We show that, under some conditions, if the distribution of the maximum of n i.i.d. variables is of the same type for two distinct values of n then the distribution is one of the three extreme value types. This is an analogue of the well known result that if the sum of two i.i.d. random variables with finite second moment is of the same type as the original distribution then the distribution is Gaussian (Kagan et al., 1973). Our result was motivated by study of the m out of n bootstrap.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bickel, P. and Sakov, A., On the choice of m in the m out of n bootstrap in estimation problems, In preparation, 2002.
Castro, L., deHaan, L., and Temido, M., “Rarely observed sample maxima,” Theory Probability and its Applications 45, 779–782, (2000).
David, H., Order statistics, (2nd edition), John Wiley & Sons, 1981.
Embrechts, P., Klüppelberg, C., and Mikosch, T., Modelling Extremal Events, Springer, 1997.
Feller, W., An introduction to probability theory and its applications, Vol. II (2nd edition), John Wiley & Sons, 1966.
Kagan, A., Linnik, Y., and Rao, C., Characterization problems in Mathematical Statistics, Wiley, 1973.
Reiss, R., Approximate distributions of order, Springer, 1989.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bickel, P.J., Sakov, A. Equality of Types for the Distribution of the Maximum for Two Values of n Implies Extreme Value Type. Extremes 5, 45–53 (2002). https://doi.org/10.1023/A:1020930008948
Issue Date:
DOI: https://doi.org/10.1023/A:1020930008948