A Note on The Backfitting Estimation of Additive Models

The additive model is one of the most popular semiparametric models. The back-fitting estimation (Buja, Hastie and Tibshirani, 1989, Ann. Statist.) for the model is intuitively easy to understand and theoretically most efficient (Opsomer and Rup-pert, 1997, Ann. Statist.); its implementation is equivalent to solving simple linear equations. However, convergence of the algorithm is very difficult to investigate and is still unsolved. For bivariate additive models, Opsomer and Ruppert (1997, Ann. Statist.) proved the convergence under a very strong condition and conjectured that a much weaker condition is sufficient. In this short note, we show that a weak condi-tion can guarantee the convergence of the backfitting estimation algorithm when the Nadaraya-Watson kernel smoothing is used. Key words: additive model; backfitting algorithm; convergence of algorithm; kernel smoothing.