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 theta and a truncation parameter gamma as a nuisance parameter, it is known that the maximum likelihood estimators (MLEs) (theta) over cap (gamma)(ML) and (theta) over cap (ML) of theta for known gamma and unknown gamma, respectively, and the maximum conditional likelihood estimator (theta) over cap (MCL) of theta are asymptotically equivalent. In this paper, the stochastic expansions of (theta) over cap (gamma)(ML) (theta) over cap (ML) and (theta) over cap (MCL) are derived, and their second-order asymptotic variances are obtained. The second-order asymptotic loss of a bias-adjusted MLE (theta) over cap (ML)* relative to (theta) over cap (gamma)(ML) is also given, and (theta) over cap (ML)* and (theta) over cap (MCL) are shown to be second-order asymptotically equivalent. Further, some examples are given.