In this paper we propose methods for inference of the geometric features of a
multivariate density. Our approach uses multiscale tests for the monotonicity
of the density at arbitrary points in arbitrary directions. In particular, a
significance test for a mode at a specific point is constructed. Moreover, we
develop multiscale methods for identifying regions of monotonicity and a
general procedure for detecting the modes of a multivariate density. It is is
shown that the latter method localizes the modes with an effectively optimal
rate. The theoretical results are illustrated by means of a simulation study
and a data example. The new method is applied to and motivated by the
determination and verification of the position of high-energy sources from
X-ray observations by the Swift satellite which is important for a
multiwavelength analysis of objects such as Active Galactic Nuclei.Comment: Keywords and Phrases: multiple tests, modes, multivariate density,
X-ray astronomy AMS Subject Classification: 62G07, 62G10, 62G2
