## An Edgeworth expansion for symmetric finite population statistics.(English)Zbl 1010.62017

From the paper: Given a set $${\mathcal X}=\{x_1, \dots,x_N)$$, let $$(X_1, \dots,X_N)$$ be a random permutation of the ordered set $$(x_2,\dots, x_N)$$. We assume that the random permutation is uniformly distributed over the class of permutations. Let $$T=t(X_1, \dots,X_n)$$ denote a symmetric statistic of the first $$n$$ observations $$X_1,\dots, X_n$$, where $$n<N$$. That is, $$t$$ is a real function defined on the class of subsets $$\{x_{i_1}, \dots, x_{i_n}\} \subset {\mathcal X}$$ of size $$n$$ and we assume that $$t(x_{i_1}, \dots, x_{i_n})$$ is invariant under permutations of its arguments. Since $$X_1,\dots, X_n$$ represents a sample drawn without replacement from the population $${\mathcal X}$$, we call $$T$$ a symmetric finite population statistic.
Assuming that the linear part of Hoeffding’s decomposition of $$T$$ is nondegenerate we construct a one term Edgeworth expansion for the distribution function of $$T$$ and prove the validity of the expansion with the remainder $$O(1/n^*)$$ as $$n^*\to \infty$$, where $$n^*=\min \{n,N-n\}$$.

### MSC:

 62E20 Asymptotic distribution theory in statistics 62D05 Sampling theory, sample surveys 60F05 Central limit and other weak theorems
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### References:

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