Smoothed Versions of Statistical Functionals from a Finite Population

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  • Motoyama Hitoshi
    The Graduate School of Economics, Hitotsubashi University, 2-1 Naka, Kunitachi-shi, Tokyo, Japan and Statistical Information Institute for Consulting and Analysis, 3-6 Kanda Jinbocho, Chiyoda-ku, Tokyo, Japan. This research is supported in part by the Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology 15653014, 17330046 and in part a grant from the 21st Century COE Program ``Research Unit for Statistical Analysis in Social Sciences'' at Hitotsubashi University.
  • Takahashi Hajime
    The Graduate School of Economics, Hitotsubashi University, 2-1 Naka, Kunitachi-shi, Tokyo, Japan. This research is supported in part by the Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology 18203013.

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We will consider the central limit theorem for the smoothed version of statistical functionals in a finite population. For the infinite population, Reeds (1976) and Fernholz (1983) discuss the problem under the conditions of Hadamard differentiability of the statistical functionals and derive Taylor type expansions. Lindeberg-Feller's central limit theorem is applied to the leading term, and controlling the remainder terms, the central limit theorem for the statistical functionals are proved. We will modify Fernholz's method and apply it to the finite population with smoothed empirical distribution functions, and we will also obtain Taylor type expansions. We then apply the Erdös-Rényi central limit theorem to the leading linear term to obtain the central limit theorem. We will also obtain sufficient conditions for the central limit theorem, both for the smoothed influence function, and the original non-smoothed versions. Some Monte Carlo simulation results are also included.

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