The Annals of Statistics

Edgeworth Series for Lattice Distributions

John E. Kolassa and Peter McCullagh

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This paper investigates the use of Edgeworth expansions for approximating the distribution function of the normalized sum of $n$ independent and identically distributed lattice-valued random variables. We prove that the continuity-corrected Edgeworth series, using Sheppard-adjusted cumulants, is accurate to the same order in $n$ as the usual Edgeworth approximation for continuous random variables. Finally, as a partial justification of the Sheppard adjustments, it is shown that if a continuous random variable $Y$ is rounded into a discrete part $D$ and a truncation error $U$, such that $Y = D + U$, then under suitable limiting conditions the truncation error is approximately uniformly distributed and independent of $Y$, but not independent of $D$.

Article information

Ann. Statist., Volume 18, Number 2 (1990), 981-985.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62E20: Asymptotic distribution theory
Secondary: 60F05: Central limit and other weak theorems

Cumulant Edgeworth series lattice distribution rounding error Sheppard correction


Kolassa, John E.; McCullagh, Peter. Edgeworth Series for Lattice Distributions. Ann. Statist. 18 (1990), no. 2, 981--985. doi:10.1214/aos/1176347637.

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