# A Semiparametric Maximum Likelihood Estimator

https://doi.org/0012-9682(199707)65:4<933:ASMLE>2.0.CO;2-0
p. 933-963

hunrong Ai

This paper presents a procedure for analyzing a model in which the parameter vector has two parts: a finite-dimensional component $\theta$ and a nonparametric component $\lambda$. The procedure does not require parametric modeling of $\lambda$ but assumes that the true density of the data satisfies an index restriction. The idea is to construct a parametric model passing through the true model and to estimate $\theta$ by setting the score for the parametric model to zero. The score is estimated nonparametrically and the estimator is shown to be $\sqrt N$ consistent and asymptotically normal. The estimator is then shown to attain the semiparametric efficiency bound characterized in Begun et al. (1983) for multivariate nonlinear regression, simultaneous equations, partially specified regression, index regression, censored regression, switching regression, and disequilibrium models in which the error densities are unknown.