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Bahadur efficiency of the maximum likelihood estimator and one-step estimator for quasi-arithmetic means of the Cauchy distribution

Author

Listed:
  • Yuichi Akaoka

    (Shinshu University
    Gunma bank)

  • Kazuki Okamura

    (Shizuoka University)

  • Yoshiki Otobe

    (Shinshu University)

Abstract

Some quasi-arithmetic means of random variables easily give unbiased strongly consistent closed-form estimators of the joint of the location and scale parameters of the Cauchy distribution. The one-step estimators of those quasi-arithmetic means of the Cauchy distribution are considered. We establish the Bahadur efficiency of the maximum likelihood estimator and the one-step estimators. We also show that the rate of the convergence of the mean-squared errors achieves the Cramér–Rao bound. Our results are also applicable to the circular Cauchy distribution .

Suggested Citation

  • Yuichi Akaoka & Kazuki Okamura & Yoshiki Otobe, 2022. "Bahadur efficiency of the maximum likelihood estimator and one-step estimator for quasi-arithmetic means of the Cauchy distribution," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 74(5), pages 895-923, October.
    • Handle: RePEc:spr:aistmt:v:74:y:2022:i:5:d:10.1007_s10463-021-00818-y
    DOI: 10.1007/s10463-021-00818-y
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    References listed on IDEAS

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    1. Muneya Matsui & Akimichi Takemura, 2005. "Empirical characteristic function approach to goodness-of-fit tests for the Cauchy distribution with parameters estimated by MLE or EISE," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 57(1), pages 183-199, March.
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    6. Nora Gürtler & Norbert Henze, 2000. "Goodness-of-Fit Tests for the Cauchy Distribution Based on the Empirical Characteristic Function," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(2), pages 267-286, June.
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