Under mild misspecifications of model assumptions, maximum likelihood estimates often remain consistent and asymptotically normal. Asymptotic normality will often hold for the signed root of the likelihood ratio statistic and the score statistic as well. However, standard estimates of asymptotic variance are usually inconsistent. This occurs when Bartlett's second identity fails. In the manner of McCullagh and Tibshirani, a variance correction may be used to adjust the profile likelihood so this identity obtains. The resulting likelihood yields the robust versions of the signed root, Wald and score statistic suggested by Kent and Royall.
Assuming model correctness, asymptotic expansions for the first three cumulants of each robust statistic are derived. It is seen that bias and skewness are not severely affected by using a robust statistic. An invariant expression derived for the asymptotic relative efficiency of a robust method allows assessment in numerous examples considered. Even for moderately large sample sizes, losses in efficiency are significant, making the misuse of a robust variance estimate potentially costly. Computer algebra is used in many of the calculations reported in this paper.
"A robust adjustment of the profile likelihood." Ann. Statist. 24 (1) 336 - 352, February 1996. https://doi.org/10.1214/aos/1033066212