Semiparametric estimation of a class of generalized linear models without smoothing

In a generalized linear model, the mean of the response variable is a possibly non-linear function of a linear combination of explanatory variables. When the nonlinear function is unknown and is estimated nonparametrically from the data, these models are known as single index models. Using the relation of generalized linear models with the exponential family model, this paper shows how to use a modified version of the empirical cumulant generating function to estimate the linear function of the explanatory variables with no need of smoothing techniques. The resulting estimator is consistent and normally distributed. Extensive simulations, partially reported here, show that the method works in practice. The method can also be seen as complementary to existing fully nonparametric methods. In fact, it can provide an initial value that can be used to fine tune a nonparametric estimator of the link function in the first step of the estimation.