Exact Probability Bounds under Moment-matching Restrictions

TL;DR

This paper investigates the maximum deviation from normality for distributions matching a certain number of moments, revealing that the previously assumed bounds are not always attainable and providing explicit solutions for even-moment cases.

Contribution

It clarifies the attainability of probability bounds under moment-matching restrictions and develops explicit solutions for symmetric distributions with finite mass points.

Findings

Bound is not attained when the number of matched even moments is odd.

Explicit symmetric distribution solutions are provided for even-moment cases.

The bounds for odd-moment cases are established as limits of the even-moment solutions.

Abstract

Lindsay and Basak (2000) posed the 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 c.d.f.'s, 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.