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Maximizing Probability Bounds Under Moment-Matching Restrictions

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  • Stephen Portnoy

Abstract

The problem of characterizing a distribution by its moments dates to work by Chebyshev in the mid-nineteenth century. There are clear (and close) connections with characteristic functions, moment spaces, quadrature, and other very classical mathematical pursuits. Lindsay and Basak posed the specific question of how far from normality could a distribution be if it matches k normal moments. They provided a bound on the maximal difference in cdfs, and implied that these bounds were attained. It will be shown here that in fact the bound is not attained if the number of even moments matched is odd. An explicit solution is developed as a symmetric distribution with a finite number of mass points when the number of even moments matched is even, and this bound for the even case is shown to hold as an explicit limit for the subsequent odd case. As Lindsay noted, the discrepancies can be sizable even for a moderate number of matched moments. Some comments on implications are proffered.

Suggested Citation

  • Stephen Portnoy, 2015. "Maximizing Probability Bounds Under Moment-Matching Restrictions," The American Statistician, Taylor & Francis Journals, vol. 69(1), pages 41-44, February.
    • Handle: RePEc:taf:amstat:v:69:y:2015:i:1:p:41-44
    DOI: 10.1080/00031305.2014.992960
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    References listed on IDEAS

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    1. Bruce Lindsay & Ramani Pilla & Prasanta Basak, 2000. "Moment-Based Approximations of Distributions Using Mixtures: Theory and Applications," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(2), pages 215-230, June.
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