The size of error is investigated when the log-likelihood of multivariate generalized linear models is approximated by a quadratic function. The nonquadratic tail is characterized by analyzing the cubic part of the log-likelihood. In a local analysis simple bounds for that part can be expressed in terms of expectations of the related random variables for arbitrary sample size $N$. Additionally global error bounds are given for the univariate case.
Christian Kredler. "Behaviour of Third Order Terms in Quadratic Approximations of LR-Statistics in Multivariate Generalized Linear Models." Ann. Statist. 14 (1) 326 - 335, March, 1986. https://doi.org/10.1214/aos/1176349859