On the layered nearest neighbour estimate, the bagged nearest neighbour estimate and the random forest method in regression and classification

Let X1,…,Xn be identically distributed random vectors in RdRd, independently drawn according to some probability density. An observation Xi is said to be a layered nearest neighbour (LNN) of a point x if the hyperrectangle defined by x and Xi contains no other data points. We first establish consistency results on Ln(x), the number of LNN of x. Then, given a sample (X,Y),(X1,Y1),…,(Xn,Yn) of independent identically distributed random vectors from Rd×RRd×R, one may estimate the regression function r(x)=E[Y|X=x] by the LNN estimate rn(x), defined as an average over the YiYi’s corresponding to those Xi which are LNN of x. Under mild conditions on rr, we establish the consistency of E|rn(x)−r(x)|p towards 00 as n→∞n→∞, for almost all x and all p≥1p≥1, and discuss the links between rnrn and the random forest estimates of Breiman (2001) [8]. We finally show the universal consistency of the bagged (bootstrap-aggregated) nearest neighbour method for regression and classification.