zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
An analysis of Bayesian inference for nonparametric regression. (English) Zbl 0778.62003
Summary: The observation model $y\sb i=\beta(i/n)+\varepsilon\sb i$, $1\le i\le n$, is considered, where the $\varepsilon$’s are i.i.d. with mean zero and variance $\sigma\sp 2$ 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 $\Vert\delta\Vert\sp 2$ when $\Vert\cdot\Vert$ 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 $n\to\infty$ with prior probability 1. A related continuous time signal estimation problem is also studied.

MSC:
62A01Foundations and philosophical topics in statistics
62G07Density estimation
62G15Nonparametric tolerance and confidence regions
62E20Asymptotic distribution theory in statistics
62M99Inference from stochastic processes
WorldCat.org
Full Text: DOI