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 \(n^*=\min \{n,N-n\}\).


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