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Large deviation theory for non-regular location shift family

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Abstract

We apply non-regular extensions of the large deviation theory to non-regular location shift families. Our calculation contains the location shift families generated by Beta distribution, Weibull distribution, and Gamma distribution. We point out the optimal estimator depends on the choice of our criterion in the non-regular case. The limits of relative Rényi entropies play an important role in our derivation.

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References

  • Akahira M. (1996) Loss of information of a statistic for a family of non-regular distributions. Annals of Institute of Statistical Mathematics 48(2): 349–364

    Article  MathSciNet  MATH  Google Scholar 

  • Akahira, M., Takeuchi, K. (1995). Non-regular statistical estimation. Lecture Notes in Statistics (Vol. 107). New York: Springer.

  • Amari S. (1984) Non-regular probability family and Finsler geometry (in Japanese). RIMS koukyuroku 6: 27

    Google Scholar 

  • Bahadur R.R. (1960) Asymptotic efficiency of tests and estimates. Sankhyā 22: 229–252

    MathSciNet  MATH  Google Scholar 

  • Bahadur R.R. (1967) Rates of convergence of estimates and test statistics. Annals of Mathematical Statistics 38: 303–324

    Article  MathSciNet  MATH  Google Scholar 

  • Bahadur, R. R. (1971). Some limit theorems in statistics. In Regional conference series in applied mathematics (Vol. 4). Philadelphia: SIAM.

  • Fu J.C. (1973) On a theorem of Bahadur on the rate of convergence of point estimator. Annals of Statistics 1: 745–749

    Article  MathSciNet  MATH  Google Scholar 

  • Hayashi, M. (2002a). Two quantum analogues of Fisher information from a large deviation viewpoint of quantum estimation. Journal of Physics. A:Mathematical and General, 35, 7689–7727. Also appeared as Chap. 28 of Asymptotic theory of quantum statistical inference, M. Hayashi (ed.), 2005.

  • Hayashi, M. (2002b). Limiting behavior of relative Rényi entropy in a non-regular location shift family. Annals of Institute of Statistical Mathematics. Latest version: http://arxiv.org/abs/math/0212077. doi:10.1007/s10463-008-0182-4.

  • Hayashi, M. (2008). Two non-regular extensions of large deviation bound. Special Issue—Recent Advances in Statistical Inference. Communications in Statistics: Theory and Methods (accepted). Latest version: http://arxiv.org/abs/math/0604197.

  • Ibragimov I.A., Has’minskii R.Z. (1981) Statistical estimation. Springer, New York

    MATH  Google Scholar 

  • Nagaoka, H. (1992). On the relation between Kullback divergence and Fisher information—from classical systems to quantum systems. In Proceedings of Joint Mini-workshop for data compression theory and fundamental open problems in information theory (pp. 63–72). Originally written in Japanese. Also appeared as Chap. 27 of Asymptotic theory of quantum statistical inference, M. Hayashi (ed.), 2005.

  • Nagaoka, H. (1994). Two quantum analogues of the large deviation Cramér-Rao inequality. In Proceedings of 1994 IEEE International Symposium on Information Theory (p. 118).

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  1. Graduate School of Information Sciences, Tohoku University, Aoba-ku, Sendai, 980-8579, Japan

    Masahito Hayashi

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  1. Masahito Hayashi
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Correspondence to Masahito Hayashi.

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Hayashi, M. Large deviation theory for non-regular location shift family. Ann Inst Stat Math 63, 689–716 (2011). https://doi.org/10.1007/s10463-009-0254-0

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  • DOI: https://doi.org/10.1007/s10463-009-0254-0

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