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The problem of estimating a conditional density is considered. Given a collection of partitions, we propose a procedure that selects from the data the best partition among that collection and then provides the best piecewise polynomial estimator built on that partition. The observations are not supposed to be independent but only β-mixing; in particular, our study includes the estimation of the transition density of a Markov chain. For a well-chosen collection of possibly irregular partitions,doi:10.1214/11-ejs653 fatcat:743ymhjyt5awbfmwcqv3ludbk4