Estimating generalized semiparametric additive models using parameter cascading

Elimination of nuisance parameters is a central but difficult problem in statistical inference. We propose the parameter cascading method to estimate statistical models that involve nuisance parameters, structural parameters, and complexity parameters. The parameter cascading method has several unique aspects. First, we consider functional relationships between parameters, quantitatively described using analytical derivatives. These functional relationships can be explicit or implicit, and in the latter case the Implicit Function Theorem is applied to obtain the required derivatives. Second, we can express the gradients and Hessian matrices analytically, which is essential for fast and stable computation. Third, we develop the unconditional variance estimates for parameters, which include the uncertainty coming from other parameter estimates. The parameter cascading method is demonstrated by estimating generalized semiparametric additive models (GSAMs), where the response variable is allowed to be from any distribution. The practical necessity and empirical performance of the parameter cascading method are illustrated through a simulation study, and two applied example, one on finding the effect of air pollution on public health, and the other on the management of a retirement fund. The results demonstrate that the parameter cascading method is a good alternative to traditional methods.