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

For a truncated exponential family of distributions with a natural parameter θ and a truncation parameter γ as a nuisance parameter, it is known that the maximum likelihood estimators (MLEs) θ^γML and θ^ML of θ for known γ and unknown γ, respectively, and the maximum conditional likelihood estimator θ^MCL of θ are asymptotically equivalent. In this paper, the stochastic expansions of θ^γML, θ^ML and θ^MCL are derived, and their second-order asymptotic variances are obtained. The second-order asymptotic loss of a bias-adjusted MLE θ^∗ML relative to θ^γML is also given, and θ^∗ML and θ^MCL are shown to be second-order asymptotically equivalent. Further, some examples are given