## The Annals of Statistics

- Ann. Statist.
- Volume 21, Number 2 (1993), 903-923.

### An Analysis of Bayesian Inference for Nonparametric Regression

#### Abstract

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.

#### Article information

**Source**

Ann. Statist. Volume 21, Number 2 (1993), 903-923.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aos/1176349157

**JSTOR**

links.jstor.org

**Digital Object Identifier**

doi:10.1214/aos/1176349157

**Mathematical Reviews number (MathSciNet)**

MR1232525

**Zentralblatt MATH identifier**

0778.62003

**Subjects**

Primary: 62A15

Secondary: 62G15: Tolerance and confidence regions 62J99: None of the above, but in this section 62E20: Asymptotic distribution theory 62M99: None of the above, but in this section 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx]

**Keywords**

Bayesian inference nonparametric regression confidence regions signal extraction smoothing splines

#### Citation

Cox, Dennis D. An Analysis of Bayesian Inference for Nonparametric Regression. Ann. Statist. 21 (1993), no. 2, 903--923. doi:10.1214/aos/1176349157. http://projecteuclid.org/euclid.aos/1176349157.