Abstract
We study the class on non-parametric deformed statistical models where the deformed exponential has linear growth at infinity and is sub-exponential at zero. This class generalizes the class introduced by N.J. Newton. We discuss the convexity and regularity of the normalization operator, the form of the deformed statistical divergences and their convex duality, the properties of the escort densities, and the affine manifold structure of the statistical bundle
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References
Amari, S.: Dual connections on the Hilbert bundles of statistical models. In: Geometrization of Statistical Theory (Lancaster, 1987), pp. 123–151. ULDM Publ. (1987)
Amari, S., Nagaoka, H.: Methods of Information Geometry. American Mathematical Society, Providence (2000). Translated from the 1993 Japanese original by Daishi Harada
Ambrosetti, A., Prodi, G.: A Primer of Nonlinear Analysis. Cambridge Studies in Advanced Mathematics, vol. 34. Cambridge University Press, Cambridge (1993)
Appell, J., Zabrejko, P.P.: Nonlinear Superposition Operators. Cambridge Tracts in Mathematics, vol. 95. Cambridge University Press, Cambridge (1990). https://doi.org/10.1017/CBO9780511897450
Ay, N., Jost, J., Lê, H.V., Schwachhöfer, L.: Information Geometry. Springer, Berlin (2017)
Dieudonné, J.: Foundations of Modern Analysis. Academic Press, New York (1960)
Ekeland, I., Témam, R.: Convex Analysis and Variational Problems. Classics in Applied Mathematics, vol. 28, English edn. Society for Industrial and Applied Mathematics (SIAM) (1999). https://doi.org/10.1137/1.9781611971088. Translated from the French
Gibilisco, P., Pistone, G.: Connections on non-parametric statistical manifolds by Orlicz space geometry. IDAQP 1(2), 325–347 (1998)
Kaniadakis, G.: Non-linear kinetics underlying generalized statistics. Phys. A 296(3–4), 405–425 (2001)
Kaniadakis, G.: Statistical mechanics in the context of special relativity. Phys. Rev. E 66, 056, 125 1–17 (2002)
Kaniadakis, G.: Statistical mechanics in the context of special relativity. ii. Phys. Rev. E 72(3), 036,108 (2005). https://doi.org/10.1103/PhysRevE.72.036108
Kass, R.E., Vos, P.W.: Geometrical Foundations of Asymptotic Inference. Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, New York (1997). https://doi.org/10.1002/9781118165980. A Wiley-Interscience Publication
Landau, L.D., Lifshits, E.M.: Course of Theoretical Physics. Statistical Physics, vol. V, 3rd edn. Butterworth-Heinemann, Oxford (1980)
Loaiza, G., Quiceno, H.R.: A \(q\)-exponential statistical Banach manifold. J. Math. Anal. Appl. 398(2), 466–476 (2013). https://doi.org/10.1016/j.jmaa.2012.08.046
Malliavin, P.: Integration and Probability. Graduate Texts in Mathematics, vol. 157. Springer-Verlag (1995). With the collaboration of Hlne Airault, Leslie Kay and Grard Letac. Edited and translated from the French by Kay, With a foreword by Mark Pinsky
Montrucchio, L., Pistone, G.: Deformed exponential bundle: the linear growth case. In: Nielsen, F., Barbaresco, F. (eds.) Geometric Science of Information. LNCS, vol. 10589, pp. 239–246. Springer (2017). Proceedings of the Third International Conference, GSI 2017, Paris, France, November 7–9
Naudts, J.: Generalised Thermostatistics. Springer-Verlag London Ltd. (2011). https://doi.org/10.1007/978-0-85729-355-8
Newton, N.J.: An infinite-dimensional statistical manifold modelled on Hilbert space. J. Funct. Anal. 263(6), 1661–1681 (2012). https://doi.org/10.1016/j.jfa.2012.06.007
Newton, N.J.: Infinite-dimensional statistical manifolds based on a balanced chart. Bernoulli 22(2), 711–731 (2016). https://doi.org/10.3150/14-BEJ673
Pistone, G.: \(\kappa \)-exponential models from the geometrical viewpoint. Eur. Phys. J. B Condens. Matter Phys. 71(1), 29–37 (2009). https://doi.org/10.1140/epjb/e2009-00154-y
Pistone, G.: Nonparametric information geometry. In: Nielsen, F., Barbaresco, F. (eds.) Geometric Science of Information. Lecture Notes in Computer Science, vol. 8085, pp. 5–36. Springer, Heidelberg (2013). Proceedings First International Conference, GSI 2013 Paris, France, August 28–30
Pistone, G., Rogantin, M.: The exponential statistical manifold: mean parameters, orthogonality and space transformations. Bernoulli 5(4), 721–760 (1999)
Pistone, G., Sempi, C.: An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one. Ann. Stat. 23(5), 1543–1561 (1995)
Shima, H.: The Geometry of Hessian Structures. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2007). https://doi.org/10.1142/9789812707536
Tsallis, C.: Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys. 52(1–2), 479–487 (1988)
Vigelis, R.F., Cavalcante, C.C.: On \(\phi \)-families of probability distributions. J. Theor. Probab. 26, 870–884 (2013)
Acknowledgements
The Authors wish to thank the anonymous referees whose comments have led to a considerable improvement of the paper. L. Montrucchio is Honorary Fellow of the Collegio Carlo Alberto Foundation. G. Pistone is a member of GNAMPA-INdAM and acknowledges the support of de Castro Statistics and Collegio Carlo Alberto.
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Montrucchio, L., Pistone, G. (2019). A Class of Non-parametric Deformed Exponential Statistical Models. In: Nielsen, F. (eds) Geometric Structures of Information. Signals and Communication Technology. Springer, Cham. https://doi.org/10.1007/978-3-030-02520-5_2
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DOI: https://doi.org/10.1007/978-3-030-02520-5_2
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