Before we proceed with proving the lemma, we first establish some shorthand notation. We are now in a position to prove (A.2). Combining this with (A.4) shows that max Rearranging the resulting expression yields the desired identity. Fact 1 is a special case of the following lemma. Moreover, we can express the variance d V AR ( g ( X) | X 2 t) in a similar form, viz., d V AR ( g ( X) | X 2 t) = 1 N (t) With these specifications fixed, the program (A.19) becomes min Hence by (A.12), we obtain the desired (A.8).

Decision trees and their ensembles are endowed with a rich set of diagnostic tools for ranking and screening input variables in a predictive model. One of the most commonly used in practice is the Mean Decrease in Impurity (MDI), which calculates an importance score for a variable by summing the weighted impurity reductions over all non-terminal nodes split with that variable. Despite the widespread use of tree based variable importance measures such as MDI, pinning down their theoretical properties has been challenging and therefore largely unexplored. To address this gap between theory and practice, we derive rigorous finite sample performance guarantees for variable ranking and selection in nonparametric models with MDI for a single-level CART decision tree (decision stump). We find that the marginal signal strength of each variable and ambient dimensionality can be considerably weaker and higher, respectively, than state-of-the-art nonparametric variable selection methods. Furthermore, unlike previous marginal screening methods that attempt to directly estimate each marginal projection via a truncated basis expansion, the fitted model used here is a simple, parsimonious decision stump, thereby eliminating the need for tuning the number of basis terms. Thus, surprisingly, even though decision stumps are highly inaccurate for estimation purposes, they can still be used to perform consistent model selection.

Decision trees with binary splits are popularly constructed using Classification and Regression Trees (CART) methodology. For regression models, this approach recursively divides the data into two near-homogenous daughter nodes according to a split point that maximizes the reduction in sum of squares error (the impurity) along a particular variable. This paper aims to study the statistical properties of regression trees constructed with CART methodology. In doing so, we find that the training error is governed by the Pearson correlation between the optimal decision stump and response data in each node, which we bound by constructing a prior distribution on the split points and solving a nonlinear optimization problem. We leverage this connection between the training error and Pearson correlation to show that CART with cost-complexity pruning achieves an optimal complexity/goodnessof-fit tradeoff when the depth scales with the logarithm of the sample size. Data dependent quantities, which adapt to the dimensionality and latent structure of the regression model, are seen to govern the rates of convergence of the prediction error.