bounds for a combinatorial central limit theorem with involutions
نویسنده
چکیده
Let E = ((eij))n×n be a fixed array of real numbers such that eij = eji, eii = 0 for 1 ≤ i, j ≤ n. Let the symmetric group be denoted by Sn and the collection of involutions with no fixed points by Πn, that is, Πn = {π ∈ Sn : π 2 = id, π(i) 6= i∀i}. For π uniformly chosen from Πn, let YE = Pn i=1 eiπ(i) and W = (YE − μE)/σE where μE = E(YE) and σ 2 E = Var(YE). Denoting by FW and Φ the distribution functions of W and a N (0, 1) variate respectively, we bound ||FW − Φ||p for 1 ≤ p ≤ ∞ using Stein’s method and zero bias transformations. The resulting bound obtained is the product of a third moment type quantity multiplied by an explicit constant, and in particular for p = ∞ is of the same form as the one obtained by Bolthausen for Hoeffding’s combinatorial central limit theorem when π is chosen uniformly from Sn. The approximation developed here for involutions has applications in testing whether there is a significant degree of similarity in certain matched pairs experiments.
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