2,371 research outputs found
Extrema of locally stationary Gaussian fields on growing manifolds
We consider a class of non-homogeneous, continuous, centered Gaussian random
fields where denotes
a rescaled smooth manifold, i.e. and study
the limit behavior of the extreme values of these Gaussian random fields when
tends to zero, which means that the manifold is growing. Our main result
can be thought of as a generalization of a classical result of Bickel and
Rosenblatt (1973a), and also of results by Mikhaleva and Piterbarg (1997).Comment: 28 pages, 1 figur
Theoretical Analysis of Nonparametric Filament Estimation
This paper provides a rigorous study of the nonparametric estimation of
filaments or ridge lines of a probability density . Points on the filament
are considered as local extrema of the density when traversing the support of
along the integral curve driven by the vector field of second eigenvectors
of the Hessian of . We `parametrize' points on the filaments by such
integral curves, and thus both the estimation of integral curves and of
filaments will be considered via a plug-in method using kernel density
estimation. We establish rates of convergence and asymptotic distribution
results for the estimation of both the integral curves and the filaments. The
main theoretical result establishes the asymptotic distribution of the uniform
deviation of the estimated filament from its theoretical counterpart. This
result utilizes the extreme value behavior of non-stationary Gaussian processes
indexed by manifolds as .Comment: 55 pages, 1 figur
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