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Self-consistency and a generalized principal subspace theorem

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  • Tarpey, Thaddeus
  • Loperfido, Nicola

Abstract

Principal subspace theorems deal with the problem of finding subspaces supporting optimal approximations of multivariate distributions. The optimality criterion considered in this paper is the minimization of the mean squared distance between the given distribution and an approximating distribution, subject to some constraints. Statistical applications include, but are not limited to, cluster analysis, principal components analysis and projection pursuit. Most principal subspace theorems deal with elliptical distributions or with mixtures of spherical distributions. We generalize these results using the notion of self-consistency. We also show their connections with the skew-normal distribution and projection pursuit techniques. We also discuss their implications, with special focus on principal points and self-consistent points. Finally, we access the practical relevance of the theoretical results by means of several simulation studies.

Suggested Citation

  • Tarpey, Thaddeus & Loperfido, Nicola, 2015. "Self-consistency and a generalized principal subspace theorem," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 27-37.
    • Handle: RePEc:eee:jmvana:v:133:y:2015:i:c:p:27-37
    DOI: 10.1016/j.jmva.2014.08.012
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    Cited by:

    1. Loperfido, Nicola, 2015. "Vector-valued skewness for model-based clustering," Statistics & Probability Letters, Elsevier, vol. 99(C), pages 230-237.
    2. Shushi, Tomer, 2019. "The Minkowski length of a spherical random vector," Statistics & Probability Letters, Elsevier, vol. 153(C), pages 104-107.
    3. 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.
    4. 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.
    5. 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).

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