QUASI-BAYESIAN ANALYSIS OF NONPARAMETRIC INSTRUMENTAL VARIABLES MODELS

• 成果类型: Article
• 署名作者: Kato, Kengo
• 署名单位: University of Tokyo
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/13-AOS1150
• 发表日期: 2013
• 页码: 2359-2390
• 关键词: posterior distributions; convergence-rates; Inverse problems; Asymptotic Normality; exponential-families; inference; regression; contraction; complexity; number

摘要

This paper aims at developing a quasi-Bayesian analysis of the nonparametric instrumental variables model, with a focus on the asymptotic properties of quasi-posterior distributions. In this paper, instead of assuming a distributional assumption on the data generating process, we consider a quasi-likelihood induced from the conditional moment restriction, and put priors on the function-valued parameter. We call the resulting posterior quasi-posterior, which corresponds to Gibbs posterior in the literature. Here we focus on priors constructed on slowly growing finite-dimensional sieves. We derive rates of contraction and a nonparametric Bernstein-von Mises type result for the quasi-posterior distribution, and rates of convergence for the quasi-Bayes estimator defined by the posterior expectation. We show that, with priors suitably chosen, the quasi-posterior distribution (the quasi-Bayes estimator) attains the minimax optimal rate of contraction (convergence, resp.). These results greatly sharpen the previous related work.