2003, Volume 65, Pt. 4 , 807--820

**A CHARACTERIZATION OF THE
MARSHALL-OLKIN DEPENDENCE FUNCTION AND A GOODNESS-OF-FIT TEST**

By

MICHAEL FALK, Universit\"{a}t W\"{u}rzburg, Germany and ROLF-DIETER REISS, Universit\"{a}t Siegen, Germany

SUMMARY. A bivariate extreme value distribution function (EV) \textit{G} with reversed exponential marginals can be represented as $G(x,y)=\exp((x+y)D(y/(x+y)))$, $x,y\le 0$, where $D: [0,1]\rightarrow[1/2,1]$ is the dependence function. The corresponding generalized Pareto distribution function (GP) is $W(x,y)=1+(x+y)D(y/(x+y))$ if $(x+y)D(y/(x+y))\ge -1$. We discuss some applications of the Pickands coordinates of $(x,y)$ including estimators of the dependence function $D$ and their asymptotic distributions in the GP model and in the EV model. The concept of bivariate $\delta$-neighbourhoods of GPs is introduced and together with a characterization of the Marshall-Olkin dependence function in terms of a differential equation for $D(z)$ at $z=1/2$, we define a goodness-of-fit test for the $\delta$-neighbourhood of the Marshall-Olkin GP.

*AMS (1991) subject classification. *Primary 62H12; secondary 62H05.

*Key words and phrases. *Bivariate extreme value distribution,
bivariate generalized Pareto distribution, Pickands representation, dependence
function, Marshall-Olkin dependence function, Pickands coordinates, bivariate
$\delta$-neighbourhoods, goodness-of-fit test for Mar\-shall-Olkin.