Fitting the additive model by recursion on dimension

We consider estimation for the homoscedastic additive model for multiple regression. A recursion is proposed in Opsomer (1999), and independently by the authors, for obtaining the estimators that solve the normal equations given by Hastie and Tibshirani (1990). The recursion can be exploited to obtain the asymptotic bias and variance expressions of the estimators for any p > 2 (Opsomer 1999) using repeated application of Opsomer and Ruppert (1997). Opsomer and Ruppert (1997) provide asymptotic bias and variance for the estimators when p = 2. Opsomer (1999) also uses the recursion to provide sufficient conditions for convergence of the backfitting algorithm to a unique solution of the normal equations. However, since explicit expressions for the solution to the normal equations are not given, he states, “The lemma does not provide a practical way of evaluating the existence and uniqueness of the backfitting estimators … ”. In this paper, explicit expressions for the estimators are derived. The explicit solution requires inverses of n × n matrices to solve the np × np system of normal equations. These matrix inverses are feasible to implement for moderate sample sizes and can be used in place of the backfitting algorithm.