## The Annals of Statistics

- Ann. Statist.
- Volume 18, Number 3 (1990), 1070-1090.

### A Decomposition for the Likelihood Ratio Statistic and the Bartlett Correction--A Bayesian Argument

Peter J. Bickel and J. K. Ghosh

#### Abstract

Let $l(\theta) = n^{-1} \log p(x, \theta)$ be the log likelihood of an $n$-dimensional $X$ under a $p$-dimensional $\theta$. Let $\hat{\theta}_j$ be the mle under $H_j: \theta^1 = \theta^1_0, \ldots, \theta^j = \theta^j_0$ and $\hat{\theta}_0$ be the unrestricted mle. Define $T_j$ as $\lbrack 2n\{l(\hat{\theta}_{j - 1}) - l(\hat{\theta}_j)\}\rbrack^{1/2} \operatorname{sgn}(\hat{\theta}^j_{j - 1} - \theta^j_0).$ Let $T = (T_1, \ldots, T_p)$. Then under regularity conditions, the following theorem is proved: Under $\theta = \theta_0, T$ is asymptotically $N(n^{-1/2}a_0 + n^{-1}a, J + n^{-1}\sum) + O(n^{-3/2})$ where $J$ is the identity matrix. The result is proved by first establishing an analogous result when $\theta$ is random and then making the prior converge to a degenerate distribution. The existence of the Bartlett correction to order $n^{-3/2}$ follows from the theorem. We show that an Edgeworth expansion with error $O(n^{-2})$ for $T$ involves only polynomials of degree less than or equal to 3 and hence verify rigorously Lawley's (1956) result giving the order of the error in the Bartlett correction as $O(n^{-2})$.

#### Article information

**Source**

Ann. Statist., Volume 18, Number 3 (1990), 1070-1090.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176347740

**Digital Object Identifier**

doi:10.1214/aos/1176347740

**Mathematical Reviews number (MathSciNet)**

MR1062699

**Zentralblatt MATH identifier**

0727.62035

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62F05: Asymptotic properties of tests

Secondary: 62F15: Bayesian inference

**Keywords**

Bartlett correction signed log likelihood ratio statistic Bernstein-von Mises theorem

#### Citation

Bickel, Peter J.; Ghosh, J. K. A Decomposition for the Likelihood Ratio Statistic and the Bartlett Correction--A Bayesian Argument. Ann. Statist. 18 (1990), no. 3, 1070--1090. doi:10.1214/aos/1176347740. https://projecteuclid.org/euclid.aos/1176347740