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

  • Masafumi AkahiraEmail author


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.


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



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.


  1. Akahira, M. (1986). The structure of asymptotic deficiency of estimators. Queens Papers in Pure and Applied Mathematics, 75. Kingston: Queen’s University Press.Google Scholar
  2. Akahira, M., Ohyauchi, N. (2012). The asymptotic expansion of the maximum likelihood estimator for a truncated exponential family of distributions (In Japanese). Kôkyûroku: RIMS (Research Institute for Mathematical Sciences, Kyoto University), 1804, 188–192.Google Scholar
  3. Akahira, M., Takeuchi, K. (1982). On asymptotic deficiencies of estimators in pooled samples in the presence of nuisance parameters. Statistics and Decisions, 1, (pp. 17–38). Also included in. (2003). Joint Statistical Papers of Akahira and Takeuchi (pp. 199–220). New Jersey: World Scientific.Google Scholar
  4. Akahira, M, Hashimoto, S., Koike, K., Ohyauchi, N. (2014). Second order asymptotic comparison of the MLE and MCLE for a two-sided truncated exponential family of distributions. Mathematical Research Note 2014–001, Institute of Mathematics, University of Tsukuba. To appear in Communications in Statistics—Theory and Methods.Google Scholar
  5. Andersen, E. B. (1970). Asymptotic properties of conditional maximum likelihood estimators. Journal of the Royal Statistical Society, Series B, 32, 283–301.MathSciNetzbMATHGoogle Scholar
  6. Bar-Lev, S. K. (1984). Large sample properties of the MLE and MCLE for the natural parameter of a truncated exponential family. Annals of the Institute of Statistical Mathematics, 36(Part A), 217–222.Google Scholar
  7. Bar-Lev, S. K., Reiser, B. (1983). A note on maximum conditional likelihood estimation. Sankhyā, Series A, 45, 300–302.Google Scholar
  8. Barndorff-Nielsen, O. E. (1978). Information and Exponential Families in Statistical Theory. New York: Wiley.zbMATHGoogle Scholar
  9. Barndorff-Nielsen, O. E., Cox, D. R. (1994). Inference and Asymptotics. London: Chapman & Hall.Google Scholar
  10. Basu, D. (1977). On the elimination of nuisance parameters. Journal of the American Statistical Association, 72, 355–366.MathSciNetCrossRefzbMATHGoogle Scholar
  11. Cox, D. R., Reid, N. (1987). Parameter orthogonality and approximate conditional inference (with discussion). Journal of the Royal Statistical Society, Series B, 49, 1–39.Google Scholar
  12. Ferguson, H. (1992). Asymptotic properties of a conditional maximum likelihood estimator. Canadian Journal of Statistics, 20, 63–75.MathSciNetCrossRefzbMATHGoogle Scholar
  13. Hodges, J. L., Lehmann, E. L. (1970). Deficiency. Annals of Mathematical Statistics, 41, 783–801.Google Scholar
  14. Huque, F., Katti, S. K. (1976). A note on maximum conditional estimators. Sankhyā, Series B, 38, 1–13.Google Scholar
  15. Liang, K.-Y. (1984). The asymptotic efficiency of conditional likelihood methods. Biometrika, 71, 305–313.MathSciNetCrossRefzbMATHGoogle Scholar
  16. Quesenberry, C. P. (1975). Transforming samples from truncation parameter distributions to uniformity. Communications in Statistics, 4, 1149–1155.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2015

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of TsukubaTsukubaJapan

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