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Identifiability of Asymmetric Circular and Cylindrical Distributions

Author

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  • Yoichi Miyata

    (Takasaki City University of Economics)

  • Takayuki Shiohama

    (Nanzan University)

  • Toshihiro Abe

    (Hosei University)

Abstract

Identifiability of statistical models is a fundamental and essential condition that is required to prove the consistency of maximum likelihood estimators. The identifiability of the skew families of distributions on the circle and cylinder for estimating model parameters has not been fully investigated in the literature. In this paper, a new method combining the trigonometric moments and the simultaneous Diophantine approximation is proposed to prove the identifiability of asymmetric circular and cylindrical distributions. Using this method, we prove the identifiability of general sine-skewed circular distributions, including the sine-skewed von Mises and sine-skewed wrapped Cauchy distributions, and that of a Möbius transformed cardioid distribution, which can be regarded as asymmetric distributions on the unit circle. In addition, we prove the identifiability of two cylindrical distributions wherein both marginal distributions of a circular random variable are the sine-skewed wrapped Cauchy distribution, and conditional distributions of a random variable on the non-negative real line given the circular random variable are a Weibull distribution and a generalized Pareto-type distribution, respectively.

Suggested Citation

  • Yoichi Miyata & Takayuki Shiohama & Toshihiro Abe, 2023. "Identifiability of Asymmetric Circular and Cylindrical Distributions," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(2), pages 1431-1451, August.
    • Handle: RePEc:spr:sankha:v:85:y:2023:i:2:d:10.1007_s13171-022-00294-3
    DOI: 10.1007/s13171-022-00294-3
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    References listed on IDEAS

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    1. Umbach, Dale & Jammalamadaka, S. Rao, 2009. "Building asymmetry into circular distributions," Statistics & Probability Letters, Elsevier, vol. 79(5), pages 659-663, March.
    2. Toshihiro Abe & Arthur Pewsey, 2011. "Sine-skewed circular distributions," Statistical Papers, Springer, vol. 52(3), pages 683-707, August.
    3. Rothenberg, Thomas J, 1971. "Identification in Parametric Models," Econometrica, Econometric Society, vol. 39(3), pages 577-591, May.
    4. Abe, Toshihiro & Ley, Christophe, 2017. "A tractable, parsimonious and flexible model for cylindrical data, with applications," Econometrics and Statistics, Elsevier, vol. 4(C), pages 91-104.
    5. M. C. Jones & Arthur Pewsey, 2012. "Inverse Batschelet Distributions for Circular Data," Biometrics, The International Biometric Society, vol. 68(1), pages 183-193, March.
    6. Kato, Shogo & Jones, M. C., 2010. "A Family of Distributions on the Circle With Links to, and Applications Arising From, Möbius Transformation," Journal of the American Statistical Association, American Statistical Association, vol. 105(489), pages 249-262.
    7. Yoichi Miyata & Takayuki Shiohama & Toshihiro Abe, 2020. "Estimation of finite mixture models of skew-symmetric circular distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 83(8), pages 895-922, November.
    8. Ley, Christophe & Verdebout, Thomas, 2017. "Skew-rotationally-symmetric distributions and related efficient inferential procedures," Journal of Multivariate Analysis, Elsevier, vol. 159(C), pages 67-81.
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