We introduce a new family of distributions to approximate P(WβA) for Aβ{...,β2,β1,0,1,2,...} and W a sum of independent
integer-valued random variables ΞΎ1β, ΞΎ2β, ...,ΞΎnβ with finite
second moments, where, with large probability, W is not concentrated on a
lattice of span greater than 1. The well-known Berry--Esseen theorem states
that, for Z a normal random variable with mean E(W) and variance
Var(W), P(ZβA) provides a good approximation
to P(WβA) for A of the form (ββ,x]. However, for more
general A, such as the set of all even numbers, the normal approximation
becomes unsatisfactory and it is desirable to have an appropriate discrete,
nonnormal distribution which approximates W in total variation, and a
discrete version of the Berry--Esseen theorem to bound the error. In this
paper, using the concept of zero biasing for discrete random variables (cf.
Goldstein and Reinert [J. Theoret. Probab. 18 (2005) 237--260]), we introduce a
new family of discrete distributions and provide a discrete version of the
Berry--Esseen theorem showing how members of the family approximate the
distribution of a sum W of integer-valued variables in total variation.Comment: Published at http://dx.doi.org/10.1214/009117906000000250 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org