Hoeffding-ANOVA decompositions for symmetric statistics of exchangeable observations

Abstract

Consider a (possibly infinite) exchangeable sequence X = {Xn: 1 ≤ n 0, Xn(α,c) consists of the first n instants of a generalized Pólya urn sequence. For every choice of α(·) and c, the Hoeffding-ANOVA decomposition of a symmetric and square integrable statistic T(Xn(α,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 Hoeffding [Ann. Math. Statist. 19 (1948) 293-325] for i.i.d. variables, Zhao and Chen [Acta Math. Appl. Sinica 6 (1990) 263-272] and Bloznelis and Götze [Ann. Statist. 29 (2001) 353-365 and Ann. Probab. 30 (2002) 1238-1265] 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.