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October 2013 Quasi-Bayesian analysis of nonparametric instrumental variables models
Kengo Kato
Ann. Statist. 41(5): 2359-2390 (October 2013). DOI: 10.1214/13-AOS1150

Abstract

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.

Citation

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Kengo Kato. "Quasi-Bayesian analysis of nonparametric instrumental variables models." Ann. Statist. 41 (5) 2359 - 2390, October 2013. https://doi.org/10.1214/13-AOS1150

Information

Published: October 2013
First available in Project Euclid: 5 November 2013

zbMATH: 1281.62120
MathSciNet: MR3127869
Digital Object Identifier: 10.1214/13-AOS1150

Subjects:
Primary: 62G08 , 62G20

Keywords: asymptotic normality , inverse problem , nonparametric instrumental variables model , quasi-Bayes , rates of contraction

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 5 • October 2013
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