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

  • Konstantin Eckle
  • Nicolai Bissantz
  • Holger Dette
  • Katharina Proksch
  • Sabrina Einecke


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.


Multiple tests Modes Multivariate density X-ray astronomy 



This research has made use of data obtained through the High Energy Astrophysics Science Archive Research Center Online Service, provided by the NASA/Goddard Space Flight Center. We are very grateful to a reviewer and an associate editor for their constructive comments on an earlier version of this paper. The authors would also like to thank Martina Stein, who typed parts of this manuscript with considerable technical expertise. This work has been supported in part by the Collaborative Research Center “Statistical modeling of nonlinear dynamic processes” (SFB 823, Teilprojekt C1, C4) of the German Research Foundation (DFG).


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Copyright information

© The Institute of Statistical Mathematics, Tokyo 2017

Authors and Affiliations

  • Konstantin Eckle
    • 1
  • Nicolai Bissantz
    • 1
  • Holger Dette
    • 1
  • Katharina Proksch
    • 2
  • Sabrina Einecke
    • 3
  1. 1.Fakultät für MathematikRuhr-Universität BochumBochumGermany
  2. 2.Institut für Mathematische StochastikGeorg-August-Universität GöttingenGöttingenGermany
  3. 3.Fakultät PhysikTechnische Universität DortmundDortmundGermany

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