Second Order Asymptotic Loss of the MLE of a Truncation Parameter for a Two-Sided Truncated Exponential Family of Distributions

For a one-sided truncated exponential family of distributions with a truncation parameter and a natural parameter as a nuisance parameter, it is shown by Akahira and Ohyauchi (2016) that the second order asymptotic loss of a bias-adjusted maximum likelihood estimator (MLE) of a truncation parameter for unknown natural parameter relative to a bias-adjusted MLE of a truncation parameter for known natural parameter is obtained. In this paper, in a similar way to Akahira and Ohyauchi (2016), for a two-sided truncated exponential family of distributions with a natural parameter and lower and upper truncation parameters, the stochastic expansions of the bias-adjusted MLE of an upper truncation parameter for known natural and lower truncation parameters, the bias-adjusted MLE of an upper truncation parameter for unknown natural parameter and known lower truncation parameter and the bias-adjusted MLE of an upper truncation parameter for unknown natural and lower truncation parameters are derived, their asymptotic variances are given, and the second order asymptotic losses of the MLEs of an upper truncation parameter for unknown natural parameter and known/unknown lower truncation parameter relative to the MLE of an upper truncation parameter for known natural and lower truncation parameters are also obtained. Further, some examples including an upper-truncated Pareto case are given.