Asymptotic consistency of median regression trees

Estimation of a function parameter by adaptive recursive partitioning of the covariate space is a well known and effective non-parametric statistical technique. Piecewise constant nonparametric estimate of a conditional median function based on least absolute deviations regression tree has been proposed and discussed by Breiman et al. (Classification and Regression Trees. Wadsworth, Belmont, 1984). In this article, we derive and discuss some general regularity conditions that can ensure asymptotic consistency of such an estimate. We also discuss a weighted average technique following Chaudhuri et al. (Statist. Sinica 4 (1994) 143) based on smooth weight functions that can nicely glue the discontinuous constant pieces of such an estimate to yield a smooth and consistent estimate