items that pass through this inspection process, and are still contaminated, is given by $$\label{eq1.3} {\bar X}_1 \biggl(1-\frac{n_1}{ N} \biggr)+{\bar
X}_2 \biggl(1-\frac
{n_2}{ N} \biggr) ,$$ which can be viewed as a sum of means of lattice-valued random variables where the lattice edge lengths are $e_j=1-N^{-1}n_j$ for $j=1,2$.

The quarantine inspection service aims to develop a strategy for choosing consignments, and items, to inspect. This reduces the associated costs, and minimises, to at least some extent, the number of contaminated items that cross the border. The performance of such a strategy is assessed, by the quarantine service, using a variety of statistics based on sums of binomials; (\[eq1.3\]) is just one example. Quarantine services are usually interested in providing confidence intervals as well as point estimators, and hence there is significant interest in estimating the distributions of statistics such as that at (\[eq1.3\]).

Main results {#sec2}
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Edgeworth expansions with remainder equal to $o(n^{-1/2})$ {#sec2.1}
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Let $X_{ji}$, for $1\leq i\leq n_j$ and $j=1,\ldots,k$, denote independent random variables. Assume that each $X_{ji}$ has a nondegenerate lattice distribution, depending on $j$ but not on $i$ and with maximal lattice edge width $e_j$ and finite third moment. Suppose too that $k\geq2$. Put 