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\centerline{\bf STANFORD UNIVERSITY}
\centerline{\bf DEPARTMENT OF STATISTICS}
\centerline{\big SPECIAL SEMINAR}
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\centerline{4:15 p.m., Wednesday, October 4, 2000}
\centerline{Sequoia Hall Rm. 200}
\centerline{(Cookies at 3:45 in 1st Floor Lounge)}

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\centerline{\sl Saralees Nadarajah}
\centerline{\sl University of California, Santa Barbara}
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\centerline{\bf Ordered Multivariate Extremes}
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In recent years statistical extreme value theory has matured to such an
extent to contribute usefully to the study of substantial real problems,
particularly in the area of environmental extremes. Examples include the
design of off-shore structures (Coles and Tawn, 1994) and the study of
reservoir flood safety (Anderson and Nadarajah, 1993).

A fairly commonly occurring characteristic is that the variables whose
extremes are of interest are ordered.  In hydro-meteorology one thing that
is of interest is the dependence of extreme values of $d$-hour rainfall
over a range of values of $d$.  One approach is to fit a multivariate
extreme value distribution over that range. If $X(d)$ denotes rainfall
aggregated over $d$ hours, and if $\hat d > d$ then $X(d) \leq X(\hat d) \leq
(\hat d/d) X(d)$ for all $(X(d), X(\hat d))$, so an order restriction in the
multivariate extreme value model is needed. Similar order restrictions
arise in the study of the joint distributions of large hourly mean wind
speeds and large wind gusts.

The aim of this talk is to develop multivariate extremal models and
associated statistical procedures for vector observations whose components
are subject to an order relationship.  We consider only the bivariate case.
 The results are applied to the joint analysis of rainfall extremes
corresponding to different durations.

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