Kitagawa, Toru Olea, José Luis Montiel Payne, Jonathan Velez, Amilcar
Year of Publication:
cemmap working paper No. CWP17/19
This paper examines the asymptotic behavior of the posterior distribution of a possibly nondifferentiable function g(θ), where θ is a finite-dimensional parameter of either a parametric or semiparametric model. The main assumption is that the distribution of a suitable estimator θ, its bootstrap approximation, and the Bayesian posterior for θ all agree asymptotically. It is shown that whenever g is locally Lipschitz, though not necessarily differentiable, the posterior distribution of g(θ) and the bootstrap distribution of g(θ) coincide asymptotically. One implication is that Bayesians can interpret bootstrap inference for g(θ) as approximately valid posterior inference in a large sample. Another implication—built on known results about bootstrap inconsistency—is that credible intervals for a nondifferentiable parameter g(θ) cannot be presumed to be approximately valid confidence intervals (even when this relation holds true for θ).