## The Annals of Statistics

### Asymptotic normality with small relative errors of posterior probabilities of half-spaces

#### Abstract

Let $\Theta$ be a parameter space included in a finite-dimensional Euclidean space and let $A$ be a half-space. Suppose that the maximum likelihood estimate $\theta_n$ of $\theta$ is not in $A$ (otherwise, replace $A$ by its complement) and let $\Delta$ be the maximum log likelihood (at $\theta_n$) minus the maximum log likelihood over the boundary $\partial A$. It is shown that under some conditions, uniformly over all half-spaces $A$, either the posterior probability of $A$ is asymptotic to $\Phi(-\sqrt{2\Delta}\,)$ where $\Phi$ is the standard normal distribution function, or both the posterior probability and its approximant go to 0 exponentially in $n$. Sharper approximations depending on the prior are also defined.

#### Article information

Source
Ann. Statist., Volume 30, Number 5 (2002), 1311-1344.

Dates
First available in Project Euclid: 28 October 2002

https://projecteuclid.org/euclid.aos/1035844978

Digital Object Identifier
doi:10.1214/aos/1035844978

Mathematical Reviews number (MathSciNet)
MR1936321

Zentralblatt MATH identifier
1014.62031

#### Citation

Dudley, R. M.; Haughton, D. Asymptotic normality with small relative errors of posterior probabilities of half-spaces. Ann. Statist. 30 (2002), no. 5, 1311--1344. doi:10.1214/aos/1035844978. https://projecteuclid.org/euclid.aos/1035844978

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• CAMBRIDGE, MASSACHUSETTS 02139 E-MAIL: rmd@math.mit.edu DEPARTMENT OF MATHEMATICAL SCIENCES BENTLEY COLLEGE 175 FOREST ST.
• WALTHAM, MASSACHUSETTS 02154 E-MAIL: dhaughton@bentley.edu