IJPAM: Volume 12, No. 4 (2004)

EXTREME VALUE TYPE DISTRIBUTIONS
WITH BOUNDED SUPPORT

Boris I. Kunin
Department of Mathematical Sciences
University of Alabama in Huntsville
Huntsville, AL 35899, USA
e-mail: kunin@math.uah.edu


Abstract.Underlying the derivation of the classical extreme value distributions is the group $\mathcal{A}^{+}$ of orientation preserving affine transformations of the real line. The group $\mathcal{F}^{+}$ of orientation preserving fractional linear transformations is the only Lie group of analytic transformations of the (extended) real line that contains $\mathcal{A}^{+}$. This singular place occupied by $\mathcal{F}^{+}$ is one of the reasons to consider the consequences of using $\mathcal{F}^{+}$ in place of $\mathcal{A}%
^{+}$.

It is shown that there exists a unique compactly supported four parameter distribution $G$, 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 $G$ come from $%
\mathcal{F}^{+}$, not $\mathcal{A}^{+}$. The same holds for minima.

As in the classical case, the distribution function $G$ is alternatively characterized as ``stable'', i.e., for $n=1,2,...,$ one has $G^{n}=G\circ q_{n} $ for some $q_{n}\in \mathcal{F}^{+}$ (on the support of $G$). It is shown that the sequence $q_{n}$ can be extended to a continuous, hence smooth, one parameter subgroup $q_{\exp (\sigma )}$ of $\mathcal{F}^{+}$. This allows to obtain $G$ 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
ISSN: 1311-8080
Year: 2004
Volume: 12
Issue: 4