Nonparametric Estimation and Testing of Interaction in Generalized Additive Models

Abstract

The additive model overcomes the "curse of dimensionality" in general nonparametric regression problems, in the sense that it achieves the optimal rate of convergence for a one-dimensional smoother. Meanwhile, compared to the classical linear regression model, it is more flexible in defining an arbitrary smooth functional relationship between the individual regressor and the conditional mean of the response variable Y given X. However, if the true model is not additive, the estimates may be seriously biased by assuming the additive structure. In this dissertation, generalized additive models (with a known link function) are considered when containing second order interaction terms. We present an extension of the existing marginal integration estimation approach for additive models with the identity link. The corresponding asymptotic normality of the estimators is derived for the univariate component functions and interaction functions. A test statistic for testing significance of the interaction terms is developed. We obtained the asymptotics for the test functional and local power results. Monte Carlo simulations are conducted to examine the finite sample performance of the estimation and testing procedures. We code our own local polynomial pre-smoother with fixed bandwidths and apply it in the integration method. The widely used LOESS function with fixed spans is also used as a pre-smoother. Both methods provide comparable results in estimation and are shown to work well with properly chosen smoothing parameters. With a small and moderate sample size, the implementation of the test procedure based on the asymptotics may produce inaccurate results. Hence a wild bootstrap procedure is provided to get empirical critical values for the test. The test procedure performs well in fitting the correct quantiles under the null hypothesis and shows strong power against the alternative.