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On the limiting distribution of sample central moments

Author

Listed:
  • Georgios Afendras

    (Aristotle University of Thessaloniki)

  • Nickos Papadatos

    (National and Kapodistrian University of Athens)

  • Violetta E. Piperigou

    (University of Patras)

Abstract

We investigate the limiting behavior of sample central moments, examining the special cases where the limiting (as the sample size tends to infinity) distribution is degenerate. Parent (non-degenerate) distributions with this property are called singular, and we show in this article that the singular distributions contain at most three supporting points. Moreover, using the delta-method, we show that the (second-order) limiting distribution of sample central moments from a singular distribution is either a multiple, or a difference of two multiples of independent Chi-square random variables with one degree of freedom. Finally, we present a new characterization of normality through the asymptotic independence of the sample mean and all sample central moments.

Suggested Citation

  • Georgios Afendras & Nickos Papadatos & Violetta E. Piperigou, 2020. "On the limiting distribution of sample central moments," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(2), pages 399-425, April.
    • Handle: RePEc:spr:aistmt:v:72:y:2020:i:2:d:10.1007_s10463-018-0695-4
    DOI: 10.1007/s10463-018-0695-4
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    References listed on IDEAS

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