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

M. Akahira, S. Hashimoto, K. Koike and N. Ohyauchi

Communications in Statistics - Theory and Methods, 2016, vol. 45, issue 19, 5637-5659

Abstract: For 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) θ^ML*$\hat{\theta }_{ML}^{*}$ of θ for unknown γ relative to the MLE θ^MLγ$\hat{\theta }_{ML}^{\gamma }$ of θ for known γ is given and θ^ML*$\hat{\theta }_{ML}^{*}$ and the maximum conditional likelihood estimator (MCLE) θ^MCL$\hat{\theta }_{MCL}$ 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 θ^MLγ,ν$\hat{\theta }_{ML}^{\gamma , \nu }$ of θ for known γ and ν and the MLE θ^ML$\hat{\theta }_{ML}$ and the MCLE θ^MCL$\hat{\theta }_{MCL}$ of θ for unknown γ and ν are derived, their second-order asymptotic means and variances are given, a bias-adjusted MLE θ^ML*$\hat{\theta }_{ML}^{*}$ and θ^MCL$\hat{\theta }_{MCL}$ are shown to be second-order asymptotically equivalent, and the second-order asymptotic losses of θ^ML*$\hat{\theta }_{ML}^{*}$ and θ^MCL$\hat{\theta }_{MCL}$ relative to θ^MLγ,ν$\hat{\theta }_{ML}^{\gamma , \nu }$ are also obtained. Further, some examples including an upper-truncated Pareto case are given.

Date: 2016
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DOI: 10.1080/03610926.2014.948202

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