A Comparative Study of Additive and Nonparametric Regression Estimators and Variable Selection Procedures
One of the biggest challenges in nonparametric
regression is the curse of dimensionality. Additive models are known
to overcome this problem by estimating only the individual additive
effects of each covariate. However, if the model is misspecified, the
accuracy of the estimator compared to the fully nonparametric one
is unknown. In this work the efficiency of completely nonparametric
regression estimators such as the Loess is compared to the estimators
that assume additivity in several situations, including additive and
non-additive regression scenarios. The comparison is done by
computing the oracle mean square error of the estimators with regards
to the true nonparametric regression function. Then, a backward
elimination selection procedure based on the Akaike Information
Criteria is proposed, which is computed from either the additive or
the nonparametric model. Simulations show that if the additive model
is misspecified, the percentage of time it fails to select important
variables can be higher than that of the fully nonparametric approach.
A dimension reduction step is included when nonparametric estimator
cannot be computed due to the curse of dimensionality. Finally, the
Boston housing dataset is analyzed using the proposed backward
elimination procedure and the selected variables are identified.
Additive models, local polynomial regression,
residuals, mean square error, variable selection.