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Optimal principal points estimators of multivariate distributions of location-scale and location-scale-rotation families

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  • Shun Matsuura

    (Keio University)

  • Thaddeus Tarpey

    (Wright State University)

Abstract

A set of k points that optimally summarize a distribution is called a set of k-principal points, which is a generalization of the mean from one point to multiple points and is useful especially for multivariate distributions. This paper discusses the estimation of principal points of multivariate distributions. First, an optimal estimator of principal points is derived for multivariate distributions of location-scale families. In particular, an optimal principal points estimator of a multivariate normal distribution is shown to be obtained by using principal points of a scaled multivariate t-distribution. We also study the case of multivariate location-scale-rotation families. Numerical examples are presented to compare the optimal estimators with maximum likelihood estimators.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:stpapr:v:61:y:2020:i:4:d:10.1007_s00362-018-0995-z
    DOI: 10.1007/s00362-018-0995-z
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    References listed on IDEAS

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    1. Matsuura, Shun & Kurata, Hiroshi, 2011. "Principal points of a multivariate mixture distribution," Journal of Multivariate Analysis, Elsevier, vol. 102(2), pages 213-224, February.
    2. Thaddeus Tarpey, 1997. "Estimating principal points of univariate distributions," Journal of Applied Statistics, Taylor & Francis Journals, vol. 24(5), pages 499-512.
    3. Tarpey, Thaddeus & Petkova, Eva & Lu, Yimeng & Govindarajulu, Usha, 2010. "Optimal Partitioning for Linear Mixed Effects Models: Applications to Identifying Placebo Responders," Journal of the American Statistical Association, American Statistical Association, vol. 105(491), pages 968-977.
    4. 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.
    5. Thaddeus Tarpey, 2007. "A parametric k-means algorithm," Computational Statistics, Springer, vol. 22(1), pages 71-89, April.
    6. Shun Matsuura & Hiroshi Kurata, 2014. "Principal points for an allometric extension model," Statistical Papers, Springer, vol. 55(3), pages 853-870, August.
    7. Tarpey T. & Petkova E. & Ogden R.T., 2003. "Profiling Placebo Responders by Self-Consistent Partitioning of Functional Data," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 850-858, January.
    8. Tarpey, Thaddeus & Loperfido, Nicola, 2015. "Self-consistency and a generalized principal subspace theorem," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 27-37.
    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.
    10. 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.
    11. Li, Luning & Flury, Bernard, 1995. "Uniqueness of principal points for univariate distributions," Statistics & Probability Letters, Elsevier, vol. 25(4), pages 323-327, December.
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    Cited by:

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