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Information geometry on the curved q-exponential family with application to survival data analysis

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  • Zhang, Fode
  • Ng, Hon Keung Tony
  • Shi, Yimin

Abstract

Information geometry and statistical manifold have attracted wide attention in the past few decades. The Amari–Chentsov structure plays a central role both in optimization and statistical inference because of the invariance under sufficient statistics. This paper discusses the Amari–Chentsov structure on statistical manifolds induced by the curved q-exponential family with Type-I censored data. The reliability function and cumulant generating function of the family are obtained, where the cumulant generating function contains a random variable depends on the experimental design. After the construction of the statistical manifold, the geometric quantities, i.e., Fisher metric, Amari tensor and connection coefficients are investigated. We find that the affine connection is not equal to 0, which is different from the existing results. The relationship among the geometric quantities based on the q-exponential family and curved q-exponential family is discussed. The relationship between the curvature on the manifold in the normal and tangent directions is obtained. Employing the maximum likelihood and Bayesian methods, the parameter vector and its dual are estimated. The main results are illustrated with the two-parameter q-exponential distribution and a real data analysis based on a terminal human cancer data set.

Suggested Citation

  • Zhang, Fode & Ng, Hon Keung Tony & Shi, Yimin, 2018. "Information geometry on the curved q-exponential family with application to survival data analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 512(C), pages 788-802.
    • Handle: RePEc:eee:phsmap:v:512:y:2018:i:c:p:788-802
    DOI: 10.1016/j.physa.2018.08.143
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    References listed on IDEAS

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    1. Zhang, Fode & Ng, Hon Keung Tony & Shi, Yimin, 2018. "On alternative q-Weibull and q-extreme value distributions: Properties and applications," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 490(C), pages 1171-1190.
    2. Saberian, E., 2018. "On the spectrum of plasma modes in a field-free pair plasma: Dispersion and Landau damping in Tsallis statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 490(C), pages 289-299.
    3. Cafaro, Carlo, 2017. "Geometric algebra and information geometry for quantum computational software," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 470(C), pages 154-196.
    4. Kumar, Vikas, 2016. "Some results on Tsallis entropy measure and k-record values," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 462(C), pages 667-673.
    5. Amari, Shun-ichi & Ohara, Atsumi & Matsuzoe, Hiroshi, 2012. "Geometry of deformed exponential families: Invariant, dually-flat and conformal geometries," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(18), pages 4308-4319.
    6. Zhang, Fode & Shi, Yimin & Wang, Ruibing, 2017. "Geometry of the q-exponential distribution with dependent competing risks and accelerated life testing," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 468(C), pages 552-565.
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