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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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    1. Georgios Afendras & Marianthi Markatou, 2016. "Uniform integrability of the OLS estimators, and the convergence of their moments," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(4), pages 775-784, December.
    2. Yatracos, Yannis G., 2005. "Artificially Augmented Samples, Shrinkage, and Mean Squared Error Reduction," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 1168-1175, December.
    3. Stavros Kourouklis, 2012. "A New Estimator of the Variance Based on Minimizing Mean Squared Error," The American Statistician, Taylor & Francis Journals, vol. 66(4), pages 234-236, November.
    4. S. Haug & C. Klüppelberg & A. Lindner & M. Zapp, 2007. "Method of moment estimation in the COGARCH(1,1) model," Econometrics Journal, Royal Economic Society, vol. 10(2), pages 320-341, July.
    5. Stefanski L. A. & Boos D. D., 2002. "The Calculus of M-Estimation," The American Statistician, American Statistical Association, vol. 56, pages 29-38, February.
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