IJPAM: Volume 12, No. 4 (2004)
WITH BOUNDED SUPPORT
Department of Mathematical Sciences
University of Alabama in Huntsville
Huntsville, AL 35899, USA
Abstract.Underlying the derivation of the classical extreme value distributions is the group of orientation preserving affine transformations of the real line. The group of orientation preserving fractional linear transformations is the only Lie group of analytic transformations of the (extended) real line that contains . This singular place occupied by is one of the reasons to consider the consequences of using in place of .
It is shown that there exists a unique compactly supported four parameter distribution , which is a weak limit of rescaled distributions of sample maxima as the sample size tends to infinity. In contrast to the case of classical max-value distributions, the ``rescalings'' for come from , not . The same holds for minima.
As in the classical case, the distribution function is
alternatively characterized as ``stable'', i.e., for
(on the support of ). It is shown that the sequence can
be extended to a continuous, hence smooth, one parameter subgroup
. This allows to obtain
from a simple differential equation.
Received: February 27, 2004
AMS Subject Classification: 60G70, 62E20, 62G30
Key Words and Phrases: extreme value distribution, weak convergence
Source: International Journal of Pure and Applied Mathematics