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Mode-based estimation of the center of symmetry

Author

Listed:
  • José E. Chacón

    (Universidad de Extremadura, Campus Universitario de Badajoz)

  • Javier Fernández Serrano

    (Universidad Autónoma de Madrid, Ciudad Universitaria de Cantoblanco)

Abstract

In the mean-median-mode triad of univariate centrality measures, the mode has been overlooked for estimating the center of symmetry in continuous and unimodal settings. This paper expands on the connection between kernel mode estimators and M-estimators for location, bridging the gap between the nonparametrics and robust statistics communities. The variance of modal estimators is studied in terms of a bandwidth parameter, establishing conditions for an optimal solution that outperforms the household sample mean. A purely nonparametric approach is adopted, modeling heavy-tailedness through regular variation. The results lead to an estimator proposal that includes a novel one-parameter family of kernels with compact support, offering extra robustness and efficiency. The effectiveness and versatility of the new method are demonstrated in a real-world case study and a thorough simulation study, comparing favorably to traditional and more competitive alternatives. Several myths about the mode are clarified along the way, reopening the quest for flexible and efficient nonparametric estimators.

Suggested Citation

  • José E. Chacón & Javier Fernández Serrano, 2025. "Mode-based estimation of the center of symmetry," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 77(5), pages 685-717, October.
    • Handle: RePEc:spr:aistmt:v:77:y:2025:i:5:d:10.1007_s10463-025-00942-z
    DOI: 10.1007/s10463-025-00942-z
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    References listed on IDEAS

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    1. J. Chacón & J. Montanero & A. Nogales & P. Pérez, 2009. "Partial sufficiency and density estimation," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 21(8), pages 969-975.
    2. Tao Wang, 2024. "Nonlinear kernel mode‐based regression for dependent data," Journal of Time Series Analysis, Wiley Blackwell, vol. 45(2), pages 189-213, March.
    3. Purkayastha, Sumitra, 1998. "Simple proofs of two results on convolutions of unimodal distributions," Statistics & Probability Letters, Elsevier, vol. 39(2), pages 97-100, August.
    4. Joseph Romano, 1988. "Bootstrapping the mode," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 40(3), pages 565-586, September.
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