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
Full Text:

References:

 [1] Alberink, I. B. and Bentkus, V. (1999). Bounds for the concentration of asymptotically normal statistics. Acta Appl. Math. 58 11–59. · Zbl 0951.62039 [2] Aldous, D. J. (1983). Exchangeability and related topics. École d’été de Probabilités de Saint-Flour XIII . Lecture Notes in Math. 1117 . Springer, New York. · Zbl 0562.60042 [3] Arcones, M. A. and Giné, E. (1993). Limit theorems for $$U$$-processes. Ann. Probab. 21 1494–1542. JSTOR: · Zbl 0789.60031 [4] Bentkus, V., Götze, F. and van Zwet, W. R. (1997). An Edgeworth expansion for symmetric statistics. Ann. Statist. 25 851–896. · Zbl 0920.62016 [5] Blackwell, D. (1973). Discreteness of Ferguson selections. Ann. Statist. 1 356–358. · Zbl 0276.62009 [6] Blackwell, D. and MacQueen, J. (1973). Ferguson distribution via Pólya urn schemes. Ann. Statist. 1 353–355. · Zbl 0276.62010 [7] Bloznelis, M. and Götze, F. (2001). Orthogonal decomposition of finite population statistics and its applications to distributional asymptotics. Ann. Statist. 29 353–365. · Zbl 1012.62009 [8] Bloznelis, M. and Götze, F. (2002). An Edgeworth expansion for finite population statistics. Ann. Probab. 30 1238–1265. · Zbl 1010.62017 [9] Chow, Y. S. and Teicher, H. (1978). Probability Theory . Springer, New York. · Zbl 0399.60001 [10] Dudley, R. M. (1989). Real Analysis and Probability . Wadsworth and Brooks/Cole, Pacific Grove, CA. · Zbl 0686.60001 [11] Efron, B. and Stein, C. (1981). The jackknife estimate of variance. Ann. Statist. 9 586–596. JSTOR: · Zbl 0481.62035 [12] Ferguson, T. S. (1973). A Bayesian analysis of some non-parametric problems. Ann. Statist. 1 209–230. · Zbl 0255.62037 [13] Ferguson, T. S. (1974). Prior distributions on spaces of probability measures. Ann. Statist. 2 615–629. JSTOR: · Zbl 0286.62008 [14] Föllmer, H., Wu, C.-T. and Yor, M. (2000). On weak Brownian motions of arbitrary orders. Ann. Instit. H. Poincaré 36 447–487. · Zbl 0968.60069 [15] Friedrich, K. O. (1989). A Berry–Esseen bound for functions of independent random variables. Ann. Statist. 17 170–183. JSTOR: · Zbl 0671.60016 [16] Hájek, J. (1968). Asymptotic normality of simple linear rank statistics under alternatives. Ann. Math. Statist. 39 325–346. · Zbl 0187.16401 [17] Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. Ann. Math. Statist. 19 293–325. · Zbl 0032.04101 [18] Karlin, S. and Rinott, Y. (1982). Applications of ANOVA type decompositions for comparisons of conditional variance statistics including jackknife estimates. Ann. Statist. 10 485–501. JSTOR: · Zbl 0491.62036 [19] Koroljuk, V. S. and Borovskich, Yu. V. (1994). Theory of $$U$$- Statistics . Kluwer, London. · Zbl 0785.60015 [20] Mauldin, R. D., Sudderth, W. D. and Williams, S. C. (1992). Pólya trees and random distributions. Ann. Statist. 20 1203–1221. JSTOR: · Zbl 0765.62006 [21] Peccati, G. (2002a). Multiple integral representation for functionals of Dirichlet processes. Prépublication n. 748 du Laboratoire de Probabilités et Modèles Aléatoires de l’Univ. Paris VI. [22] Peccati, G. (2002b). Chaos Brownien d’espace-temps, décompositions de Hoeffding et problèmes de convergence associés. Ph.D. thesis, Univ. Paris VI. [23] Peccati, G. (2003). Hoeffding decompositions for exchangeable sequences and chaotic representation of functionals of Dirichlet processes. Comptes Rendus Mathématiques 336 845–850. · Zbl 1036.60028 [24] Pitman, J. (1996). Some developments of the Blackwell–MacQueen urn scheme. In Statistics , Probability and Game Theory : Papers in Honor of David Blackwell . IMS, Hayward, CA. [25] Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics . Wiley, New York. · Zbl 0538.62002 [26] Takemura, A. (1983). Tensor analysis of ANOVA decomposition. J. Amer. Statist. Assoc. 78 894–900. · Zbl 0535.62061 [27] Vitale, R. A. (1990). Covariances of symmetric statistics. J. Multivariate Anal. 41 14–26. · Zbl 0759.62021 [28] Zhao, L. and Chen, X. (1990). Normal approximation for finite-population $$U$$-statistics. Acta Math. Appl. Sinica 6 263–272. · Zbl 0724.60024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.