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

Author

Listed:
  • Konstantin Eckle

    (Ruhr-Universität Bochum)

  • Nicolai Bissantz

    (Ruhr-Universität Bochum)

  • Holger Dette

    (Ruhr-Universität Bochum)

  • Katharina Proksch

    (Georg-August-Universität Göttingen)

  • Sabrina Einecke

    (Technische Universität Dortmund)

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.

Suggested Citation

  • Konstantin Eckle & Nicolai Bissantz & Holger Dette & Katharina Proksch & Sabrina Einecke, 2018. "Multiscale inference for a multivariate density with applications to X-ray astronomy," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(3), pages 647-689, June.
    • Handle: RePEc:spr:aistmt:v:70:y:2018:i:3:d:10.1007_s10463-017-0605-1
    DOI: 10.1007/s10463-017-0605-1
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    References listed on IDEAS

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    1. Christopher R. Genovese & Marco Perone-Pacifico & Isabella Verdinelli & Larry Wasserman, 2016. "Non-parametric inference for density modes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(1), pages 99-126, January.
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    3. Gerber, Samuel & Potter, Kristin, 2012. "Data Analysis with the Morse-Smale Complex: The msr Package for R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 50(i02).
    4. Burman, Prabir & Polonik, Wolfgang, 2009. "Multivariate mode hunting: Data analytic tools with measures of significance," Journal of Multivariate Analysis, Elsevier, vol. 100(6), pages 1198-1218, July.
    5. Duong, Tarn & Cowling, Arianna & Koch, Inge & Wand, M.P., 2008. "Feature significance for multivariate kernel density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 52(9), pages 4225-4242, May.
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