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\centerline{\bf STANFORD UNIVERSITY}
\centerline{\bf DEPARTMENT OF STATISTICS}
\centerline{\big JOINT PROBABILITY AND STATISTICS SEMINAR}
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\centerline{4:15 p.m., Tuesday, February 19, 2002}
\centerline{Sequoia Hall Room 200}
\centerline{(Cookies at 3:45 in 1st Floor Lounge)}

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\centerline{\sl Jonathan Taylor}
\centerline{\sl Statistics Department}
\centerline{\sl Stanford University}
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\centerline{\bf A Gaussian Kinematic Formula and its relation to Euler 
characteristic densities}
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We describe some new results in approximating the distribution of the
maximum of a smooth stochastic process on a manifold, specifically Gaussian
and closely related processes built up in a natural way from i.i.d.
real-valued Gaussian processes. For this, we use the expected Euler
characteristic (EC) method and describe a Gaussian version of the classical
Kinematic Fundamental Formulae of integral geometry which is the basis of
the approximation. This result relates the Euler characteristic densities
of a class of processes to coefficients in a power series expansions of the
standard Gaussian measure of certain tubes. We give some simple
applications of the result, specifically for certain non-central
Chi-squared processes. This Gaussian KFF also shows how to use the EC
approach for certain fields with piecewise smooth level sets. As an
application, we derive the EC densities of the process given by taking the
pointwise minimum of two correlated Gaussian processes, known in the brain
imaging literature as a correlated conjunction. To validate these
approximations, we show some simulation results and sketch some details on
how we can use a (discrete) EC approach on a triangulated manifold to get
rigorous error bounds for the continuous EC approach.

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