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

Abstract

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) \(\hat{\theta }_{\mathrm{ML}}^{\gamma }\) and \(\hat{\theta }_{\mathrm{ML}}\) of \(\theta \) for known \(\gamma \) and unknown \(\gamma \), respectively, and the maximum conditional likelihood estimator \(\hat{\theta }_{\mathrm{MCL}}\) of \(\theta \) are asymptotically equivalent. In this paper, the stochastic expansions of \(\hat{\theta }_{\mathrm{ML}}^{\gamma }\), \(\hat{\theta }_{\mathrm{ML}}\) and \(\hat{\theta }_{\mathrm{MCL}}\) are derived, and their second-order asymptotic variances are obtained. The second-order asymptotic loss of a bias-adjusted MLE \(\hat{\theta }_{\mathrm{ML}}^{*}\) relative to \(\hat{\theta }_{\mathrm{ML}}^{\gamma }\) is also given, and \(\hat{\theta }_{\mathrm{ML}}^{*}\) and \(\hat{\theta }_{\mathrm{MCL}}\) are shown to be second-order asymptotically equivalent. Further, some examples are given.