= Bartlett correction for a normal variance with unknown mean
{c}
{title2=$b=11/6$}
For a sample of size $n\geq2$ from a <normal distribution> with unknown mean, the null variance <likelihood-ratio test statistic> is $W=n[-\log(U/n)+U/n-1]$, where $U\sim\chi^2_{n-1}$. With $V=(U-n)/n$, its expected Taylor terms start with $n[\mathbb EV^2/2-\mathbb EV^3/3+\mathbb EV^4/4]$. Here $\mathbb EV^2=(2n-1)/n^2$, $\mathbb EV^3=(2n-3)/n^3$, and $\mathbb EV^4=(12n^2+4n-15)/n^4$. Higher fixed-order terms contribute only $O(n^{-2})$ under valid termwise asymptotic integration. Thus $\mathbb E_0W=1+11/(6n)+O(n^{-2})$ and $W_B=W/[1+11/(6n)]$.
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