## The Annals of Statistics

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

### Edgeworth Series for Lattice Distributions

John E. Kolassa and Peter McCullagh

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176347637

**Digital Object Identifier**

doi:10.1214/aos/1176347637

**Mathematical Reviews number (MathSciNet)**

MR1056348

**Zentralblatt MATH identifier**

0703.62021

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62E20: Asymptotic distribution theory

Secondary: 60F05: Central limit and other weak theorems

**Keywords**

Cumulant Edgeworth series lattice distribution rounding error Sheppard correction

#### Citation

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