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Hoeffding-ANOVA decompositions for symmetric statistics of exchangeable observations. (English) Zbl 1055.62060

Summary: Consider a (possibly infinite) exchangeable sequence \({\mathbf X}= \{X_n:1\leq n<N\}\), where \(N\in\mathbb{N}\cup\{\infty\}\), with values in a Borel space \((A,{\mathcal A})\), and note \({\mathbf X}_n=(X_1, \dots, X_n)\). We say that \({\mathbf X}\) is Hoeffding decomposable if, for each \(n\), every square integrable, centered and symmetric statistic based on \({\mathbf X}_n\) can be written as an orthogonal sum of \(n\) \(U\)-statistics with degenerated and symmetric kernels of increasing order. The only two examples of Hoeffding decomposable sequences studied in the literature are i.i.d. random variables and extractions without replacement from a finite population.
In the first part of the paper we establish a necessary and sufficient condition for an exchangeable sequence to be Hoeffding decomposable, that is, called weak independence. We show that not every exchangeable sequence is weakly independent, and, therefore, that not every exchangeable sequence is Hoeffding decomposable.
In the second part we apply our results to a class of exchangeable and weakly independent random vectors \({\mathbf X}_n^{(\alpha,c)}= (X_1^{(\alpha,c)}, \dots,X_n^{(\alpha,c)})\) whose law is characterized by a positive and finite measure \(\alpha(\cdot)\) on \(A\) and by a real constant \(c\). For instance, if \(c=0\), \({\mathbf X}_n^{(\alpha, c)}\) is a vector of i.i.d. random variables with law \(\alpha(\cdot)/ \alpha (A)\); if \(A\) is finite, \(\alpha(\cdot)\) is integer valued and \(c=-1\), \({\mathbf X}_n^{( \alpha,c)}\) represents the first \(n\) extractions without replacement from a finite population; if \(c>0\), \({\mathbf X}_n^{(\alpha, c)}\) consists of the first \(n\) instants of a generalized Pólya urn sequence. For every choice of \(\alpha (\cdot)\) and \(c\), the Hoeffding-ANOVA decomposition of a symmetric and square integrable statistic \(T({\mathbf X}_n^{(\alpha,c)})\) is explicitly computed in terms of linear combinations of well chosen conditional expectations of \(T\).
Our formulae generalize and unify the classic results of W. Hoeffding [Ann. Math. Stat. 19, 293–325 (1948; Zbl 0032.04101)] for i.i.d. variables, L. Zhao and X. Chen [Acta Math. Appl. Sin., Engl. Ser. 6, 263–272 (1990; Zbl 0724.60024)], and M. Bloznelis and F. Götze [Ann. Stat. 29, 899–917 (2001; Zbl 1012.62009); and Ann. Probab. 30, 1238–1265 (2002; Zbl 1010.62017)] for finite population statistics. Applications are given to construct infinite “weak urn sequences” and to characterize the covariance of symmetric statistics of generalized urn sequences.

MSC:

62G99 Nonparametric inference
60G09 Exchangeability for stochastic processes
62A01 Foundations and philosophical topics in statistics
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