Semiparametric Quasi-Bayesian Inference with Dirichlet Process Priors: Application to Nonignorable Missing Responses

Quasi-Bayesian inference, in which we can use an objective function such as generalized method of moments (GMM), M-estimators, or empirical likelihoods instead of log-likelihood functions, has been studied in Bayesian statistics.However, existing quasi-Bayesian estimation methods do not incorporate Bayesian semiparametric modeling such as Dirichlet process mixtures. In this study, we propose a semiparametric quasi-Bayesian inference with Dirichlet process priors based on the method proposed by Hoshino and Igari (2017) and Igari and Hoshino (2017), which divide the objective function into likelihood function and objective function of GMM.In the proposed method, auxiliary information such as population information can be incorporated in a GMM-type function,whereas the likelihood function is expressed as infinite mixtures.In the resulting Markov chain Monte Carlo (MCMC) algorithm, the GMM-type objective function is considered in the Metropolis Hastings algorithm in the blocked Gibbs sampler. For illustrative purposes, we apply the proposed estimation method to the missing data analysis with nonignorable responses, in which the missingness depends on the dependent variable.We show the performance of our model using a simulation study.