Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### A note about Hoeffding-ANOVA decompositions for symmetric statistics of exchangeable observations

Auteur(s):

Code(s) de Classification MSC:

• 60G09 Exchangeability
• 60G99 None of the above but in this section

Résumé: For a fixed $N\geq 2$, we consider random vectors $\mathbf{X}_{N}^{\left( \alpha ,c\right) }=\left( X_{1}^{\left( \alpha ,c\right) },...,X_{N}^{\left( \alpha ,c\right) }\right)$ with values in a product space $\left( A^{N},% \mathcal{A}^{\otimes N}\right)$ and whose law is characterized by a positive and finite measure $\alpha \left( .\right)$ on $A$, and by a real constant $c$. For instance: if $c=0$, $\mathbf{X}_{N}^{\left( \alpha ,c\right) }$ is a vector of i.i.d. random variables with law $\alpha \left( .\right) /\alpha \left( A\right)$; if $A$ is finite, $\alpha \left( .\right)$ is integer valued and $c=-1$, $\mathbf{X}_{N}^{\left( \alpha ,c\right) }$ represents the first $N$ extractions without replacement from a finite population; if $c>0$, $\mathbf{X}_{N}^{\left( \alpha ,c\right) }$ consists of the first $N$ instants of a generalized P\'{o}lya urn sequence. In all cases $\mathbf{X}_{N}^{\left( \alpha ,c\right) }$ is exchangeable. For every choice of $\alpha \left( .\right)$ and $c$, the Hoeffding-ANOVA decomposition of a symmetric and square integrable statistic $T\left( \mathbf{X}_{N}^{\left( \alpha ,c\right) }\right)$ is explicitly computed in terms of linear combinations of well chosen conditional expectations of $T$. Our formulae generalize and unify the classical results of Hoeffding (1948) for i.i.d. variables, and Zhao and Chen (1990) and Bloznelis and G\"{o}tze (2001, 2002) for finite population statistics. Two applications are given: to construct finite weak urn vectors'', and to characterize the covariance of symmetric statistics of Generalized Urn Sequences.

Mots Clés: Hoeffding-ANOVA decompositions ; Urn sequences ; Exchangeability

Date: 2002-03-27

Prépublication numéro: PMA-717

Pdf file : PMA-717.pdf