Semiparametric models consist of an easily interpreted parametric
component plus a nonparametric "nuisance" component. The astounding
flexibility of these models makes them ideal for many statistical
applications, including econometrics, survival analysis and
genetics. In this talk, we consider two related Monte Carlo methods
for inference in semiparametric models.
For the first method, we consider frequentist inference for the parametric component separately from the nuisance parameter based on sampling from the posterior of the profile likelihood. It is proved that this procedure gives a first order correct approximation to the maximum likelihood estimator of the parametric component and consistent estimation of the associated efficient Fisher information, without computing derivatives or employing complicated numerical approximations. The sampler is useful, in particular, when the nuisance parameter is not estimable at the parametric rate.
For the second method, we take the output from the first method and plug it into a suitable randomly reweighted likelihood. The resulting random objective functions are then maximized over the nuisance parameter to obtain "piggyback" bootstraps of the nonparametric component. This second procedure is only applicable when the nuisance parameter converges at the parametric rate. We verify that the resulting bootstrap procedure is consistent and demonstrate the approach with simulation studies and several examples, including both survival analysis and biased sampling applications.