3 research outputs found
Nonparametric Inference For Density Modes
We derive nonparametric confidence intervals for the eigenvalues of the
Hessian at modes of a density estimate. This provides information about the
strength and shape of modes and can also be used as a significance test. We use
a data-splitting approach in which potential modes are identified using the
first half of the data and inference is done with the second half of the data.
To get valid confidence sets for the eigenvalues, we use a bootstrap based on
an elementary-symmetric-polynomial (ESP) transformation. This leads to valid
bootstrap confidence sets regardless of any multiplicities in the eigenvalues.
We also suggest a new method for bandwidth selection, namely, choosing the
bandwidth to maximize the number of significant modes. We show by example that
this method works well. Even when the true distribution is singular, and hence
does not have a density, (in which case cross validation chooses a zero
bandwidth), our method chooses a reasonable bandwidth
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