\first{5} {Some theory for additive models } {105} \second{5.1}{Introduction}{105} \second{5.2}{Estimating equations for additive models}{106} \third{5.2.1}{$L_2$ function spaces}{107} \third{5.2.2}{Penalized least-squares}{110} \third{5.2.3}{Reproducing-kernel Hilbert-spaces}{112} \second{5.3}{Solutions to the estimating equations}{114} \third{5.3.1}{Introduction}{114} \third{5.3.2}{Projection smoothers}{115} \third{5.3.3}{Semi-parametric models}{117} \third{5.3.4}{Backfitting with two smoothers}{118} \third{5.3.5}{Existence and uniqueness: $p$-smoothers}{120} \third{5.3.6}{Convergence of backfitting: $p$-smoothers}{122} \third{5.3.7}{Summary of the main results of the section}{122} \second{5.4}{Special topics}{123} \third{5.4.1}{Weighted additive models}{123} \third{5.4.2}{A modified backfitting algorithm}{124} \third{5.4.3}{Explicit solutions to the estimating equations}{126} \third{5.4.4}{Standard errors}{126} \third{5.4.5}{Degrees of freedom}{128} \third{5.4.6}{A Bayesian version of additive models}{129} \second{5.5}{Bibliographic notes}{130} \second{5.6}{Further results and exercises 5}{131}