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.
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机译：对于具有自然参数的单边截断的指数分布族。和一个截断参数。 Akahira（2013）表明，作为一个讨厌的参数，二阶渐近损失是经过偏差调整的最大似然估计器（MLE）。 M * L的对于未知。相对于MLE。 ML的。众所周知。被给予和。 M * L和最大条件似然估计器（MCLE）。 MCL是二阶渐近等效的。本文以与Akahira（2013）相似的方式，针对具有自然参数的两边截断型指数分布族。和两个截断参数。和。作为令人讨厌的东西，MLE的随机扩展。 ML的。众所周知。和。和MLE。 ML和MCLE。 MCL。对于未知。和。推导得出它们的二阶渐近均值和方差，并进行偏差调整后的MLE。 M * L和MCL被证明是二阶渐近等效的，并且二阶渐近损失为。 M * L和相对于MCL。 。，。 ML也被获得。此外，给出了包括高截断的帕累托情况的一些示例。
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