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.