A bootstrap procedure for local semiparametric density estimation amid model uncertainties

We revisit a semiparametric procedure for density estimation based on a convex combination of a nonparametric kernel density estimator and a parametric maximum likelihood estimator, with the mixing weight locally estimated by the bootstrap method. We establish the asymptotic properties of the resulting semiparametric estimator, and show that undersmoothing at the bootstrap step is necessary if the estimator is to attain a convergence rate faster than that of the kernel density estimator under a good local parametric fit. A simulation study is conducted to investigate the finite-sample performance of the procedure. Exploiting its adaptivity to the goodness of local parametric fit, we propose a double bootstrap algorithm to incorporate into the semiparametric procedure more than one parametric family, and illustrate with a numerical example the benefits gained thereof.