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Estimating principal points of univariate distributions

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  • Thaddeus Tarpey

Abstract

The term 'principal points' originated in a problem of determining 'typical' heads for the design of protection masks, as described by Flury. Two principal points in the mask example correspond to a small and a large size. Principal points are cluster means for theoretical distributions, and sample cluster means from a k -means algorithm are non-parametric estimators of principal points. This paper demonstrates that maximum likelihood estimators and semi-parametric estimators based on symmetry constraints typically perform much better than the k -means estimators. Asymptotic results on the efficiency of these estimators of two principal points for four symmetric univariate distributions are given. Simulation results are provided to examine the performance of the estimators for finite sample sizes. Finally, the different estimators of two principal points are compared using the head dimension data for the design of protection masks.

Suggested Citation

  • Thaddeus Tarpey, 1997. "Estimating principal points of univariate distributions," Journal of Applied Statistics, Taylor & Francis Journals, vol. 24(5), pages 499-512.
    • Handle: RePEc:taf:japsta:v:24:y:1997:i:5:p:499-512
    DOI: 10.1080/02664769723503
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    References listed on IDEAS

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    1. Tarpey, Thaddeus, 1994. "Two principal points of symmetric, strongly unimodal distributions," Statistics & Probability Letters, Elsevier, vol. 20(4), pages 253-257, July.
    2. Bernard D. Flury, 1993. "Estimation of Principal Points," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 42(1), pages 139-151, March.
    3. Gutti Babu & C. Rao, 1992. "Expansions for statistics involving the mean absolute deviations," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 44(2), pages 387-403, June.
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    1. Shun Matsuura & Thaddeus Tarpey, 2020. "Optimal principal points estimators of multivariate distributions of location-scale and location-scale-rotation families," Statistical Papers, Springer, vol. 61(4), pages 1629-1643, August.
    2. Matsuura, Shun & Kurata, Hiroshi, 2011. "Principal points of a multivariate mixture distribution," Journal of Multivariate Analysis, Elsevier, vol. 102(2), pages 213-224, February.
    3. Shun Matsuura, 2014. "Effectiveness of a random compound noise strategy for robust parameter design," Journal of Applied Statistics, Taylor & Francis Journals, vol. 41(9), pages 1903-1918, September.
    4. Thaddeus Tarpey, 2007. "A parametric k-means algorithm," Computational Statistics, Springer, vol. 22(1), pages 71-89, April.
    5. Long-Hao Xu & Yinan Li & Kai-Tai Fang, 2024. "The resampling method via representative points," Statistical Papers, Springer, vol. 65(6), pages 3621-3649, August.
    6. Davidov, Ori, 2005. "When is the mean self-consistent?," Journal of Multivariate Analysis, Elsevier, vol. 96(2), pages 295-310, October.
    7. Bali, Juan Lucas & Boente, Graciela, 2009. "Principal points and elliptical distributions from the multivariate setting to the functional case," Statistics & Probability Letters, Elsevier, vol. 79(17), pages 1858-1865, September.
    8. Santanu Chakraborty & Mrinal Kanti Roychowdhury & Josef Sifuentes, 2021. "High Precision Numerical Computation of Principal Points for Univariate Distributions," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(2), pages 558-584, November.
    9. Matsuura, Shun & Kurata, Hiroshi, 2010. "A principal subspace theorem for 2-principal points of general location mixtures of spherically symmetric distributions," Statistics & Probability Letters, Elsevier, vol. 80(23-24), pages 1863-1869, December.

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