Adaptive Hausdorff estimation of density level sets

Details Adaptive Hausdorff estimation of density level sets Adaptive Hausdorff estimation of density level sets

2009-08-25

arxiv.org/abs/0908.3593v1

Annals of Statistics 2009, Vol. 37, No. 5B, 2760-2782

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/08-AOS661 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of

Scientific paper

10.1214/08-AOS661

Consider the problem of estimating the $\gamma$-level set $G^*_{\gamma}=\{x:f(x)\geq\gamma\}$ of an unknown $d$-dimensional density function $f$ based on $n$ independent observations $X_1,...,X_n$ from the density. This problem has been addressed under global error criteria related to the symmetric set difference. However, in certain applications a spatially uniform mode of convergence is desirable to ensure that the estimated set is close to the target set everywhere. The Hausdorff error criterion provides this degree of uniformity and, hence, is more appropriate in such situations. It is known that the minimax optimal rate of error convergence for the Hausdorff metric is $(n/\log n)^{-1/(d+2\alpha)}$ for level sets with boundaries that have a Lipschitz functional form, where the parameter $\alpha$ characterizes the regularity of the density around the level of interest. However, the estimators proposed in previous work are nonadaptive to the density regularity and require knowledge of the parameter $\alpha$. Furthermore, previously developed estimators achieve the minimax optimal rate for rather restricted classes of sets (e.g., the boundary fragment and star-shaped sets) that effectively reduce the set estimation problem to a function estimation problem. This characterization precludes level sets with multiple connected components, which are fundamental to many applications. This paper presents a fully data-driven procedure that is adaptive to unknown regularity conditions and achieves near minimax optimal Hausdorff error control for a class of density level sets with very general shapes and multiple connected components.

Affiliated with

Also associated with

No associations

Rating

Adaptive Hausdorff estimation of density level sets does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Adaptive Hausdorff estimation of density level sets, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Adaptive Hausdorff estimation of density level sets will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-233203