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

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## 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.

## Keywords

Truncated exponential family Natural parameter Truncation parameter Maximum likelihood estimator Maximum conditional likelihood estimator Stochastic expansion Asymptotic variance Second-order asymptotic loss## Notes

### Acknowledgments

The author thanks the referees for their careful reading and valuable comments, especially one of them for pointing out the mistake of omitting certain term in the stochastic expansions of the estimators. He also thanks the associate editor for the comment.

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