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Properties of the maximum q-likelihood estimator for independent random variables

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  • Hasegawa, Yoshihiko
  • Arita, Masanori

Abstract

In the last two decades, the nonextensive statistics proposed by Tsallis have been extensively discussed in terms of the Tsallis entropy, which is a generalization of the Boltzmann–Gibbs–Shannon entropy. Within the nonextensive framework, many kinds of generalizations have been made. Suyari [H. Suyari, IEEE Trans. on Inf. Theory, 51 (2005) 753] has proposed the generalized likelihood called the q-likelihood, in which a traditional product operator is replaced by the q-product. We study properties of the maximum q-likelihood estimator (MqLE) which is a generalization of the conventional maximum likelihood estimator with the use of the q-likelihood. We discuss MqLE from the viewpoint of the minimization of the divergences in the nonextensive statistics. It has been shown that optimum parameters determined by the MqLE are nearly in agreement with those yielding the minimum distance of the divergence proposed by Rajagopal [A.K. Rajagopal, in: S. Abe, Y. Okamoto (Eds.), Nonextensive Statistical Mechanics and Its Applications, Springer, 2000, p. 99]. We here show that the consistency of MqLE is achieved along with the non-negativity of Rajagopal’s divergence. The asymptotic Gaussianity and robustness of the MqLE for independent random variables are discussed both by analytical methods and simulations.

Suggested Citation

  • Hasegawa, Yoshihiko & Arita, Masanori, 2009. "Properties of the maximum q-likelihood estimator for independent random variables," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(17), pages 3399-3412.
    • Handle: RePEc:eee:phsmap:v:388:y:2009:i:17:p:3399-3412
    DOI: 10.1016/j.physa.2009.04.026
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    Cited by:

    1. da Silva, Sérgio Luiz E.F. & Silva, R. & dos Santos Lima, Gustavo Z. & de Araújo, João M. & Corso, Gilberto, 2022. "An outlier-resistant κ-generalized approach for robust physical parameter estimation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 600(C).
    2. Xu, Wei & Liang, Yingjie & Chen, Wen & Wang, Fajie, 2020. "Recent advances of stretched Gaussian distribution underlying Hausdorff fractal distance and its applications in fitting stretched Gaussian noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 539(C).
    3. Nelson, Kenric P. & Kon, Mark A. & Umarov, Sabir R., 2019. "Use of the geometric mean as a statistic for the scale of the coupled Gaussian distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 515(C), pages 248-257.

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