 Bartlett correction for a normal variance with unknown mean

For a sample of size $n\ge 2$ from a normal distribution with unknown mean, the null variance likelihood-ratio test statistic is $W=n[-\log (U/n)+U/n-1]$, where $U\sim {\chi }_{n-1}^2$. With $V=(U-n)/n$, its expected Taylor terms start with $n[\mathbb{E}{V}^2/2-\mathbb{E}{V}^3/3+\mathbb{E}{V}^4/4]$. Here $\mathbb{E}{V}^2=(2n-1)/n^2$, $\mathbb{E}{V}^3=(2n-3)/n^3$, and $\mathbb{E}{V}^4=(12n^2+4n-15)/n^4$. Higher fixed-order terms contribute only $O(n^{-2})$ under valid termwise asymptotic integration. Thus ${\mathbb{E}}_{0}W=1+11/(6n)+O(n^{-2})$ and $W_B=W/[1+11/(6n)]$.