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Inequalities relating maximal moments to other measures of dispersion

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  • P. C. Allaart

Abstract

Let X, X1, ..., Xk be i.i.d. random variables, and for k∈ N let Dk(X) = E(X1 V ... V Xk+1) −EX be the kth centralized maximal moment. A sharp lower bound is given for D1(X) in terms of the Lévy concentration Ql(X) = supx∈ R P(X∈[x, x + l]). This inequality, which is analogous to P. Levy's concentration‐variance inequality, illustrates the fact that maximal moments are a gauge of how much spread out the underlying distribution is. It is also shown that the centralized maximal moments are increased under convolution.

Suggested Citation

  • P. C. Allaart, 2000. "Inequalities relating maximal moments to other measures of dispersion," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 54(3), pages 366-373, November.
    • Handle: RePEc:bla:stanee:v:54:y:2000:i:3:p:366-373
    DOI: 10.1111/1467-9574.00146
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