Abstract
When for a set of random variables the underlying univariate and multivariate distributions are only approximately known, they become useless even for estimating the values of the distribution function of the extremes. In some exact models, computational difficulties might arise. Both of these can be overcome in asymptotic models. The paper discusses asymptotic models with emphasis on the availability of several dependent extreme value models. For the classical model of independent and identically distributed random variables, a functional equation is deduced for the possibly asymptotic (stable) distributions of normalized extremes. The solution of the functional equation is discussed, with emphasizing the necessity of assumptions on the domain of the equation in order to obtain the classical theory of extremes. Extensions to random sample sizes, and other related characterization theorems are also mentioned.
The tail properties of the stable distributions are discussed which lead to simple classification of population distributions in terms of their asymptotic extreme value distributions.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Baringhaus, L. (1980). Eine simultane Charakterisierung der geometrischen Verteilung und der logistischen Verteilung. Metrika 27, 237–242.
Barlow, R.E. and F. Proschan (1975). Statistical theory of reliability and life testing: Probability models. Holt, Rine-Hart and Winston, New York.
Berman, S.M. (1962). Limiting distribution of the maximum term in a sequence of dependent random variables. Ann. Math. Statist. 33, 894–908.
Berman, S.M. (1964). Limit theorems for the maximum term in stationary sequences. Ann. Math. Statist. 35, 502–516.
Cohn, H. and A. Pakes (1978). A representation for the limiting random variable of a branching process with infinite mean and some related problems. J. Appl. Probability 15, 225–234.
Deheuvels, P. (1981). Univariate extreme values - theory and applications. 43rd Session, ISI, Buenos Aires.
Dziubdziela, W. and B. Kopocinski (1976). Limiting properties of the distance random variable. (in Polish). Przeglad. Stat. 23, 471–477.
Epstein, B. (1960). Elements of the theory of extreme values. Technometrics 2, 27–41.
Galambos J. (1972). On the distribution of the maximum of random variables. Ann. Math. Statist. 43, 516–521.
Galambos, J. (1975). Limit laws for mixtures with applications to asymptotic theory of extremes. Z. Wahrsch. verw. Gebiete 32, 197–207.
Galambos, J. (1978). The asymptotic theory of extreme order statistics. Wiley, New York.
Gnedenko, B.V. (1943). Sur la distribution limite du terme maximum d’une serie aleatoire. Ann. Math. 44, 423–453. Gumbel, E.J. (1958).
Statistics of extremes. Columbia University Press, New York.
Haan, L. de (1970). On regular variation and its application to the weak convergence of sample extremes. Math. Centre tracts, 32, Amsterdam.
Ibragimov, I.A. and Yu. A. Rozanov (1970). Gaussian stochastic processes. Nauka, Moscow.
Juncosa, M.L. (1949). On the distribution of the minimum in a sequence of mutually independent random variables. Duke Math. J. 16, 609–618.
Mejzler, D.G. (1956). On the problem of the limit distribution for the maximal term of a variational series (in Russian). L’vov Politechn. Inst. Naucn. Zp. ( Fiz.-Mat. ) 38, 90–109.
Mejzler, D. (1965). On a certain class of limit distributions and their domain of attraction. Trans. Amer. Math. Soc. 117, 205–236.
Mittal, Y. and D. Ylvisaker (1975). Limit distributions for the maxima of stationary Gaussian processes. Stoch. Proc. Appl. 3, 1–18.
Mucci, R. (1977). Limit theorems for extremes. Thesis for Ph.D., Temple University.
Pickands, J. III (1967). Maxima of stationary Gaussian processes. Z. Wahrsch. verw. Gebiete 7, 190–233.
Sethuraman, J. (1965). On a characterization of the three limiting types of the extreme. Sankhya A 27, 357–364.
Shimizu, R. and L. Davies (1981). General characterization theorems for the Weibull and the stable distributions. Sankhya A 43, 282–310.
Tiago de Oliveira, J. (1976). Asymptotic behavior of maxima wi periodic disturbances. Ann. Inst. Statist. Math. 28, 19–23
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1984 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Galambos, J. (1984). Asymptotics; Stable Laws for Extremes; Tail Properties. In: de Oliveira, J.T. (eds) Statistical Extremes and Applications. NATO ASI Series, vol 131. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-3069-3_3
Download citation
DOI: https://doi.org/10.1007/978-94-017-3069-3_3
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-8401-9
Online ISBN: 978-94-017-3069-3
eBook Packages: Springer Book Archive