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Approximating symmetrized estimators of scatter via balanced incomplete U-statistics

Author

Listed:
  • Lutz Dümbgen

    (University of Bern)

  • Klaus Nordhausen

    (University of Jyväskylä)

Abstract

We derive limiting distributions of symmetrized estimators of scatter. Instead of considering all $$n(n-1)/2$$ n ( n - 1 ) / 2 pairs of the n observations, we only use nd suitably chosen pairs, where $$d \ge 1$$ d ≥ 1 is substantially smaller than n. It turns out that the resulting estimators are asymptotically equivalent to the original one whenever $$d = d(n) \rightarrow \infty$$ d = d ( n ) → ∞ at arbitrarily slow speed. We also investigate the asymptotic properties for arbitrary fixed d. These considerations and numerical examples indicate that for practical purposes, moderate fixed values of d between 10 and 20 yield already estimators which are computationally feasible and rather close to the original ones.

Suggested Citation

  • Lutz Dümbgen & Klaus Nordhausen, 2024. "Approximating symmetrized estimators of scatter via balanced incomplete U-statistics," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 76(2), pages 185-207, April.
    • Handle: RePEc:spr:aistmt:v:76:y:2024:i:2:d:10.1007_s10463-023-00879-1
    DOI: 10.1007/s10463-023-00879-1
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    References listed on IDEAS

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    1. Dümbgen, Lutz & Nordhausen, Klaus & Schuhmacher, Heike, 2016. "New algorithms for M-estimation of multivariate scatter and location," Journal of Multivariate Analysis, Elsevier, vol. 144(C), pages 200-217.
    2. Sirkiä, Seija & Taskinen, Sara & Oja, Hannu, 2007. "Symmetrised M-estimators of multivariate scatter," Journal of Multivariate Analysis, Elsevier, vol. 98(8), pages 1611-1629, September.
    3. Klaus Nordhausen & David E. Tyler, 2015. "A cautionary note on robust covariance plug-in methods," Biometrika, Biometrika Trust, vol. 102(3), pages 573-588.
    4. Paindaveine, Davy, 2008. "A canonical definition of shape," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2240-2247, October.
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