Second-order asymptotic comparison of the MLE and MCLE for a two-sided truncated exponential family of distributions

For a one-sided truncated exponential family of distributions with a natural parameter. and a truncation parameter. as a nuisance parameter, it is shown by Akahira (2013) that the second-order asymptotic loss of a bias-adjusted maximum likelihood estimator (MLE). M* L of. for unknown. relative to theMLE. ML of. for known. is given and. M*L and the maximum conditional likelihood estimator (MCLE). MCL are secondorder asymptotically equivalent. In this paper, in a similarway to Akahira (2013), for a two-sided truncated exponential family of distributions with a natural parameter. and two truncation parameters. and. as nuisance ones, the stochastic expansions of the MLE. ML of. for known. and. and the MLE. ML and the MCLE. MCL of. for unknown. and. are derived, their second-order asymptotic means and variances are given, a bias-adjusted MLE. M* L and. MCL are shown to be second-order asymptotically equivalent, and the second-order asymptotic losses of. M* L and. MCL relative to. .,. ML are also obtained. Further, some examples including an upper-truncated Pareto case are given