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A large sample study of the Bayesian bootstrap. (English) Zbl 0617.62032

This paper gives the asymptotic justification of the Bayesian bootstrap. First, through a series of lemmas, a theorem is proved. The result is: for almost all \(X(x_ 1,x_ 2,...,x_ n)\) where \(x_ 1,...,x_ n\) are i.i.d., \(D_ n\), a posterior Dirichlet process and a bootstrap empirical process \(F^*\), centered and rescaled, can be approximated by a Kiefer process in absolute deviation with rate \(O(n^{-1/4}(\log \log n)^{1/4}\) (log n)\({}^{1/2}).\)
Next, this approximation is applied to derive a large sample Bayesian bootstrap band for F, an unknown arbitrary distribution function of X. Then large sample theory for the Bayesian bootstrapping of mean and variance is discussed.
Again, the same approximation is used to show that the smoothed Bayesian bootstrap density approximates a smoothed posterior Dirichlet process density. Finally, large sample probability bands are constructed for the smoothed density and rate function.
Reviewer: G.S.Lingappaiah

MSC:

62F15 Bayesian inference
62E20 Asymptotic distribution theory in statistics
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