Second-order asymptotic loss of the MLE of a truncation parameter for a truncated exponential family of distributions

For a truncated exponential family of distributions with a truncation parameter and a natural parameter as a nuisance parameter, the stochastic expansions of bias-adjusted maximum likelihood estimators and of when is known and when is unknown, respectively, are derived. The second-order asymptotic loss of is also obtained through their asymptotic variances. Further, some examples are given.