SUFFICIENT CONDITIONS FOR UNIQUENESS OF PRINCIPAL POINTS OF SYMMETRIC UNIVARIATE DISTRIBUTIONS

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  • SHIMIZU Nobuo
    Division of Systems and Information Engineering, Hokkaido University
  • MIZUTA Masahiro
    Center for Information and Multimedia Studies, Hokkaido University
  • SATO Yoshiharu
    Division of Systems and Information Engineering, Hokkaido University

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  • Principal Pointsの対称性に関する定理について
  • Principal Points ノ タイショウセイ ニ カンスル テイリ ニ ツイテ

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In this paper, we discuss properties of uniqueness of principal points of symmetric univariate distributions. Li & Flury(1995) have proposed a sufficient condition that k principal points of symmetric univariate distributions are symmetry with respect to the expectations. We point out some mistakes in the deriving process of the sufficient condition by Li & Flury(1995), and derive the sufficient condition of uniqueness of k principal points of symmetric univariate distributions using a sufficient condition for the uniqueness of best L_2 approximation by piecewise polynomials with variable breakpoints by Chow(1982). We also derive another sufficient condition of uniqueness of k principal points of symmetric univariate distributions using a sufficient condition for the uniqueness of optimum quantizing of univariate random variables with mean-square error criterion by Trushkin(1982).

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