On the limiting distribution of sample central moments

Georgios Afendras (), Nickos Papadatos () and Violetta E. Piperigou ()
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Georgios Afendras: Aristotle University of Thessaloniki
Nickos Papadatos: National and Kapodistrian University of Athens
Violetta E. Piperigou: University of Patras

Annals of the Institute of Statistical Mathematics, 2020, vol. 72, issue 2, No 3, 399-425

Abstract: 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.

Keywords: Sample central moments; Singular distributions; Second-order approximation; Characterization of normality; Delta-method (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10463-018-0695-4

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