IDEAS home Printed from https://ideas.repec.org/a/taf/japsta/v24y1997i5p499-512.html
   My bibliography  Save this article

Estimating principal points of univariate distributions

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.tandfonline.com/doi/abs/10.1080/02664769723503
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1080/02664769723503?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. 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.
    3. 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).
    4. 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.
    5. 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.
    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. 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.
    8. 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.
    9. Thaddeus Tarpey, 2007. "A parametric k-means algorithm," Computational Statistics, Springer, vol. 22(1), pages 71-89, April.
    10. Li, Luning & Flury, Bernard, 1995. "Uniqueness of principal points for univariate distributions," Statistics & Probability Letters, Elsevier, vol. 25(4), pages 323-327, December.
    11. 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.
    12. Carina Gerstenberger & Daniel Vogel, 2015. "On the efficiency of Gini’s mean difference," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 24(4), pages 569-596, November.
    13. Kamila Tureckova, 2015. "Income Inequality By Method Of Non-Weighted Average Absolute Deviation: Case Study Of Central And Eastern European Countries*," Equilibrium. Quarterly Journal of Economics and Economic Policy, Institute of Economic Research, vol. 10(4), pages 99-110, December.
    14. Marcel, Bräutigam & Marie, Kratz, 2018. "On the Dependence between Quantiles and Dispersion Estimators," ESSEC Working Papers WP1807, ESSEC Research Center, ESSEC Business School.
    15. 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.
    16. 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.
    17. Shun Matsuura & Hiroshi Kurata, 2014. "Principal points for an allometric extension model," Statistical Papers, Springer, vol. 55(3), pages 853-870, August.
    18. Tarpey, Thaddeus, 2000. "Parallel Principal Axes," Journal of Multivariate Analysis, Elsevier, vol. 75(2), pages 295-313, November.
    19. Tarpey, Thaddeus & Loperfido, Nicola, 2015. "Self-consistency and a generalized principal subspace theorem," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 27-37.
    20. 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.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:taf:japsta:v:24:y:1997:i:5:p:499-512. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Longhurst (email available below). General contact details of provider: http://www.tandfonline.com/CJAS20 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.