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

Abstract

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