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Multiscale inference for a multivariate density with applications to X-ray astronomy

Authors
  • Eckle, Konstantin1
  • Bissantz, Nicolai1
  • Dette, Holger1
  • Proksch, Katharina2
  • Einecke, Sabrina3
  • 1 Ruhr-Universität Bochum, Fakultät für Mathematik, Bochum, 44780, Germany , Bochum (Germany)
  • 2 Georg-August-Universität Göttingen, Institut für Mathematische Stochastik, Göttingen, 37077, Germany , Göttingen (Germany)
  • 3 Technische Universität Dortmund, Fakultät Physik, Dortmund, 44221, Germany , Dortmund (Germany)
Type
Published Article
Journal
Annals of the Institute of Statistical Mathematics
Publisher
Springer Japan
Publication Date
Feb 24, 2017
Volume
70
Issue
3
Pages
647–689
Identifiers
DOI: 10.1007/s10463-017-0605-1
Source
Springer Nature
Keywords
License
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Abstract

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 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.

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