Spline-based semiparametric estimation of partially linear Poisson regression with single-index models

Epidemiological studies have shown that the high levels of air pollution are associated with the increased mortality. To further characterise the health effects of air pollutants, we propose a spline-based partially linear Poisson single-index model to study the relationship of multi-dimensional air pollution exposure to mortality. B-splines are used to approximate the unknown regression function. A modified Fisher scoring method is applied to simultaneously estimate the linear coefficients and the regression function. The estimator of the regression function is consistent with a better than cubic root convergence rate and the estimators of regression parameters are asymptotically normal and efficient. Also a simple and consistent variance estimation approach based on least-squares method is proposed. An extensive Monte Carlo study is conducted to evaluate the finite sample performance of the proposed spline approach. The method is illustrated using data from an epidemiological study of ambient fine particles.