Polynomial Spline Estimation for A Generalized Additive Coefficient Model.

We study a semiparametric generalized additive coefficient model, in which linear predictors in the conventional generalized linear models is generalized to unknown functions depending on certain covariates, and approximate the nonparametric functions by using polynomial spline. The asymptotic expansion with optimal rates of convergence for the estimators of the nonparametric part is established. Semiparametric generalized likelihood ratio test is also proposed to check if a nonparametric coefficient can be simplified as a parametric one. A conditional bootstrap version is suggested to approximate the distribution of the test under the null hypothesis. Extensive Monte Carlo simulation studies are conducted to examine the finite sample performance of the proposed methods. We further apply the proposed model and methods to a data set from a human visceral Leishmaniasis (HVL) study conduced in Brazil from 1994 to 1997. Numerical results outperform the traditional generalized linear model and the proposed generalized additive coefficient model is preferable.