Journal
BIOMETRIKA
Volume 98, Issue 3, Pages 567-582Publisher
OXFORD UNIV PRESS
DOI: 10.1093/biomet/asr030
Keywords
Density estimation; Dirichlet process mixture; Empirical Bayes; Filtering algorithm; Marginal likelihood; Martingale; Mixed effects model; Multiple testing; Profile likelihood
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Predictive recursion is an accurate and computationally efficient algorithm for nonparametric estimation of mixing densities in mixture models. In semiparametric mixture models, however, the algorithm fails to account for any uncertainty in the additional unknown structural parameter. As an alternative to existing profile likelihood methods, we treat predictive recursion as a filter approximation by fitting a fully Bayes model, whereby an approximate marginal likelihood of the structural parameter emerges and can be used for inference. We call this the predictive recursion marginal likelihood. Convergence properties of predictive recursion under model misspecification also lead to an attractive construction of this new procedure. We show pointwise convergence of a normalized version of this marginal likelihood function. Simulations compare the performance of this new approach with that of existing profile likelihood methods and with Dirichlet process mixtures in density estimation. Mixed-effects models and an empirical Bayes multiple testing application in time series analysis are also considered.
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