= Solution
A <Bartlett correction> improves calibration of a regular <likelihood-ratio test statistic> $W=2\{\ell(\widehat\theta)-\ell(\widehat\theta_0)\}$, whose null limit is $\chi_q^2$ for $q$ tested restrictions. If
$$
\mathbb E_0W=q\left(1+\frac bn+O(n^{-2})\right),
$$
the corrected statistic is $\boxed{W_B=W/(1+b/n)}$. Its null mean is $q+O(n^{-2})$ and it is compared with the same limiting <chi-squared distribution>. The coefficient is model-specific, may involve nuisance parameters, and can be calculated from likelihood cumulants or a suitable null simulation; substitution must respect the asymptotic regime.
Under the regular higher-order likelihood expansions and identities that establish Bartlett correctability, this rescaling removes the leading $O(n^{-1})$ distributional error and can give $O(n^{-2})$ chi-square calibration error. Matching one moment by itself does not prove that assertion for an arbitrary statistic, and nonregular boundary or singular models need different treatment.
For illustration, testing a normal variance with unknown mean gives $W=n[-\log(U/n)+U/n-1]$ with $U\sim\chi_{n-1}^2$. The gamma logarithmic moment yields
$$
\mathbb E_0W=n\left[\log(n/2)-\operatorname{digamma}((n-1)/2)\right]-1=1+\frac{11}{6n}+O(n^{-2}).
$$
Thus the <Bartlett correction for a normal variance with unknown mean> uses $b=11/6$, not the correction for the different known-mean model. This example also distinguishes the relative coefficient $b$ from the full divisor $1+b/n$.
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