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Estimation of Principal Points

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  • Bernard D. Flury

Abstract

The k principal points of a p‐variate random vector X are defined as those points ξ1, . . ., ξk which minimize the expected squared distance between X and the nearest of the ξj. This paper reviews some of the theory of principal points and redefines them in terms of self‐consistent points. An anthropometrical problem which initiated the theoretical developments is described. Four methods of estimation, ranging from normal theory maximum likelihood to the usual k‐means algorithm in cluster analysis, are introduced and applied to the example. Finally, a leave‐one‐out method is used to assess the performance of the four methods.

Suggested Citation

  • 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.
    • Handle: RePEc:bla:jorssc:v:42:y:1993:i:1:p:139-151
    DOI: 10.2307/2347416
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    Citations

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    Cited by:

    1. Yu, Feng, 2022. "Uniqueness of principal points with respect to p-order distance for a class of univariate continuous distribution," Statistics & Probability Letters, Elsevier, vol. 183(C).
    2. 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.
    3. Li, Luning & Flury, Bernard, 1995. "Uniqueness of principal points for univariate distributions," Statistics & Probability Letters, Elsevier, vol. 25(4), pages 323-327, December.
    4. Kai-Tai Fang & Jianxin Pan, 2023. "A Review of Representative Points of Statistical Distributions and Their Applications," Mathematics, MDPI, vol. 11(13), pages 1-25, June.
    5. Tarpey, Thaddeus & Loperfido, Nicola, 2015. "Self-consistency and a generalized principal subspace theorem," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 27-37.
    6. Matsuura, Shun & Kurata, Hiroshi, 2011. "Principal points of a multivariate mixture distribution," Journal of Multivariate Analysis, Elsevier, vol. 102(2), pages 213-224, February.
    7. Thaddeus Tarpey, 1997. "Estimating principal points of univariate distributions," Journal of Applied Statistics, Taylor & Francis Journals, vol. 24(5), pages 499-512.
    8. Jiang, Jia-Jian & He, Ping & Fang, Kai-Tai, 2015. "An interesting property of the arcsine distribution and its applications," Statistics & Probability Letters, Elsevier, vol. 105(C), pages 88-95.
    9. Tarpey, Thaddeus, 2000. "Parallel Principal Axes," Journal of Multivariate Analysis, Elsevier, vol. 75(2), pages 295-313, November.
    10. 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.
    11. 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.
    12. Yinan Li & Kai-Tai Fang & Ping He & Heng Peng, 2022. "Representative Points from a Mixture of Two Normal Distributions," Mathematics, MDPI, vol. 10(21), pages 1-28, October.
    13. Long-Hao Xu & Kai-Tai Fang & Ping He, 2022. "Properties and generation of representative points of the exponential distribution," Statistical Papers, Springer, vol. 63(1), pages 197-223, February.
    14. Pötzelberger Klaus & Strasser Helmut, 2001. "Clustering And Quantization By Msp-Partitions," Statistics & Risk Modeling, De Gruyter, vol. 19(4), pages 331-372, April.
    15. Petkova Eva & Tarpey Thaddeus & Govindarajulu Usha, 2009. "Predicting Potential Placebo Effect in Drug Treated Subjects," The International Journal of Biostatistics, De Gruyter, vol. 5(1), pages 1-27, July.
    16. Yamamoto, Wataru & Shinozaki, Nobuo, 2000. "On uniqueness of two principal points for univariate location mixtures," Statistics & Probability Letters, Elsevier, vol. 46(1), pages 33-42, January.
    17. Shun Matsuura & Hiroshi Kurata, 2014. "Principal points for an allometric extension model," Statistical Papers, Springer, vol. 55(3), pages 853-870, August.
    18. 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.
    19. Thaddeus Tarpey, 2007. "A parametric k-means algorithm," Computational Statistics, Springer, vol. 22(1), pages 71-89, April.
    20. Yang, Jun & He, Ping & Fang, Kai-Tai, 2022. "Three kinds of discrete approximations of statistical multivariate distributions and their applications," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    21. Sirao Wang & Jiajuan Liang & Min Zhou & Huajun Ye, 2022. "Testing Multivariate Normality Based on F -Representative Points," Mathematics, MDPI, vol. 10(22), pages 1-22, November.
    22. 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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