Asymptotic loss of the MLE of a truncation parameter in the presence of a nuisance parameter for a one-sided truncated family of distributions

For a truncated family of distributions with a truncation parameter gamma and a parameter theta as a nuisance parameter, we derive the stochastic expansions of bias-adjusted maximum likelihood estimators gamma ML*theta and gamma ML* of gamma based on a sample of size n when theta is known and when theta is unknown, respectively. The asymptotic loss of gamma ML* relative to gamma ML*theta is obtained up to the second order, that is the order n-1. The results are a generalization of those for a one-sided truncated exponential family of distributions. Its application to truncated t-distributions is also given.