of discontinuity than cases where the limiting irrational is transcendental. These results are extended to the case of three or more different means, and also to problems where distributions are estimated using the bootstrap. The results have practical interpretation in terms of the accuracy of inference for, among other quantities, the sum or difference of binomial proportions.=1'
address:
- 'Department of Mathematics and Statistics, The University of Melbourne, VIC 3010, Australia.\'
- 'Department of Mathematics and Statistics, The University of Melbourne, VIC 3010, Australia and Department of Statistics, University of California Davis, Davis, CA 95616, USA.\'
author:
- 
- 
title: 'Normal approximation and smoothness for sums of means of lattice-valued random variables'
---

Introduction {#sec1}
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Background: The case of a single sample mean {#sec1.1}
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Let ${\hat{\theta}}$ denote a statistical estimator of an unknown quantity ${\theta}$, and assume that ${\hat{\theta}}-{\theta}$ is asymptotically normally distributed with zero mean and variance $n^{-1}{\sigma}^2$, where $n$ is a measure of the sample size from which ${\hat{\theta}}$ was computed. In particular, the statistic $T=n^{1/2}({\hat{\theta}}-{\theta})/{\sigma}$ is asymptotically normal $\mathrm{N}(0,1)$. Under additional assumptions an Edgeworth expansion of the distribution of $T$ can generally be formulated, having the form $$\label{eq1.1} P(T\leq x)=\Phi(x)+n^{-1/2}P(x) \phi(x)+o
\bigl(n^{-1/2} \bigr),$$ uniformly in 