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Predictive recursion is an accurate and computationally efficient algorithmfor nonparametric estimation of mixing densities in mixture models. Insemiparametric mixture models, however, the algorithm fails to account for anyuncertainty in the additional unknown structural parameter. As an alternativeto existing profile likelihood methods, we treat predictive recursion as afilter approximation to fitting a fully Bayes model, whereby an approximatemarginal likelihood of the structural parameter emerges and can be used forinference. We call this the predictive recursion marginal likelihood.Convergence properties of predictive recursion under model mis-specificationalso lead to an attractive construction of this new procedure. We showpointwise convergence of a normalized version of this marginal likelihoodfunction. Simulations compare the performance of this new marginal likelihoodapproach that of existing profile likelihood methods as well as Dirichletprocess mixtures in density estimation. Mixed-effects models and an empiricalBayes multiple testing application in time series analysis are also considered.

Semiparametric inference in mixture models with predictive recursion marginal likelihood