On the backfitting algorithm for additive regression models

Abstract

We analyse additive regression model fitting via the backfitting algorithm. We show that in the case of a large class of curve estimators, which includes regressograms, simple step‐by‐step formulae can be given for the back‐fitting algorithm. The result of each cycle of the algorithm may be represented succinctly in terms of a sequence of d projections in n‐dimensional space, where d is the number of design coordinates and n is sample size. It follows from our formulae that the limit of the algorithm is simply the projection of the data onto that vector space which is orthogonal to the space of all n‐vectors fixed by each of the projections. The formulae also provide the convergence rate of the algorithm, the variance of the backfitting estimator, consistency of the estimator, and the relationship of the estimator to that obtained by directly minimizing mean squared distance.