By semiparametric models we mean partially parametric models that have an infinite-dimensional component.
One example is optimal estimation of the regression parameters p, when p, = exp(x’P) is assumed but V[yi | x,] = ю, is left unspecified. The infinite-dimensional component arises because as n ^ there are infinitely many variance parameters ю,. An optimal estimator of p, called an adaptive estimator, is one that is as efficient as that when ю, is known. Delgado and Kniesner (1997) extend results for the linear regression model to count data with exponential conditional mean function, using kernel regression methods to estimate weights to be used in a second-stage nonlinear least squares regression. In their application the estimator shows little gain over specifying ю, = p,(1 + ap,), overdispersion of the NB2 form.
A second class of semiparametric models incompletely specifies the conditional mean. Leading examples are single-index models and partially linear models. Single-index models specify p, = g(x’P) where the functional form g() is left unspecified. Partially linear models specify p, = exp(x’P + g(z,)) where the functional form g() is left unspecified. In both cases root-n consistent asymptotically normal estimators of P can be obtained, without knowledge of g().