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June, 1993 An Analysis of Bayesian Inference for Nonparametric Regression
Dennis D. Cox
Ann. Statist. 21(2): 903-923 (June, 1993). DOI: 10.1214/aos/1176349157


The observation model $y_i = \beta(i/n) + \varepsilon_i, 1 \leq i \leq n$, is considered, where the $\varepsilon$'s are i.i.d. with mean zero and variance $\sigma^2$ and $\beta$ is an unknown smooth function. A Gaussian prior distribution is specified by assuming $\beta$ is the solution of a high order stochastic differential equation. The estimation error $\delta = \beta - \hat{\beta}$ is analyzed, where $\hat{\beta}$ is the posterior expectation of $\beta$. Asymptotic posterior and sampling distributional approximations are given for $\|\delta\|^2$ when $\|\cdot\|$ is one of a family of norms natural to the problem. It is shown that the frequentist coverage probability of a variety of $(1 - \alpha)$ posterior probability regions tends to be larger than $1 - \alpha$, but will be infinitely often less than any $\varepsilon > 0$ as $n \rightarrow \infty$ with prior probability 1. A related continuous time signal estimation problem is also studied.


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Dennis D. Cox. "An Analysis of Bayesian Inference for Nonparametric Regression." Ann. Statist. 21 (2) 903 - 923, June, 1993.


Published: June, 1993
First available in Project Euclid: 12 April 2007

zbMATH: 0778.62003
MathSciNet: MR1232525
Digital Object Identifier: 10.1214/aos/1176349157

Primary: 62A15
Secondary: 60G35 , 62E20 , 62G15 , 62J99 , 62M99

Keywords: Bayesian inference , Confidence regions , Nonparametric regression , signal extraction , smoothing splines

Rights: Copyright © 1993 Institute of Mathematical Statistics


Vol.21 • No. 2 • June, 1993
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