This paper provides a rigorous study of the nonparametric estimation of filaments or ridge lines of a probability density f . Points on the filament are considered as local extrema of the density when traversing the support of f along the integral curve driven by the vector field of second eigenvectors of the Hessian of f . 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 M h , h ∈ (0, 1] as h → 0.

Bandwidth selection is crucial in the kernel estimation of density level sets. Risk based on the symmetric difference between the estimated and true level sets is usually used to measure their proximity. In this paper we provide an asymptotic L p approximation to this risk, where p is characterized by the weight function in the risk. In particular the excess risk corresponds to an L 2 type of risk, and is adopted in an optimal bandwidth selection rule for nonparametric level set estimation of d-dimensional density functions (d ≥ 1). Discussion of the assumptions:1. It is known that using higher order kernels (i.e. ν ≥ 2), together with higher order smoothness assumptions can reduce the bias in kernel density estimation. But higher order kernels

This paper studies and critically discusses the construction of nonparametric confidence regions for density level sets. Methodologies based on both vertical variation and horizontal variation are considered. The investigations provide theoretical insight into the behavior of these confidence regions via large sample theory. We also discuss the geometric relationships underlying the construction of horizontal and vertical methods, and how finite sample performance of these confidence regions is influenced by geometric or topological aspects. These discussions are supported by numerical studies.

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We consider a class of non-homogeneous, continuous, centered Gaussian random fieldsh M, and study the limit behavior of the extreme values of these Gaussian random fields when h 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).