2 On Normalization Functions and ϕ-Families of Probability Distributions
35
2.4 Conclusions
In this chapter, we investigated the behavior of the normalizing function near the
boundary of the domain of the parametrization. When Condition 2.2 occurs it was
proven that given any u belonging to the Musielak–Orlicz class, we have that ψ(αu)
converges when α tends to 1 and when u does not belong to the Musielak–Orlicz
class, it follows that ψ(αu) diverges when alpha tends to 1.
Assuming that Condition 2.2 does not occur, we proved that there exists a function
u not belonging the Musielak–Orlicz class such that ψ(αu) converges when α tends
to 1. Moreover, we observe that regardless the occurence of Condition 2.2, whenever
u is in the Musielak–Orlicz class, ψ(αu) converges since α tends to 1.
These results allow a more general class of normalizing functions and describe the
behaviour of it in the boundary domain and therefore make possible the characterization of a wider class of probability distributions for several conditions associated
to the convergence of the normalizing function.
Acknowledgements This work was supported by Coordenação de Aperfeiçoamento de Pessoal de
Nível Superior - Brasil (CAPES) - Finance Code 001 and Conselho Nacional de Desenvolvimento
Científico e Tecnológico (CNPq) (Procs. 309472/2017-2 and 408609/2016-8).
Disclaimer: Views and opinions expressed are those of the authors and do not necessarily represent
official positions of their respective companies.
References
1. Andrade, L.H.F.d., Vigelis, R.F., Vieira, F.L.J., Cavalcante, C.C.: Normalization and φ -
function: Definition and Consequences, pp. 231–238. Springer International Publishing, Cham
(2017)
2. Andrade, L.H., Vieira, F.L., F.Vigelis, R., Cavalcante, C.C.: Mixture and exponential arcs on
generalized statistical manifold. Entropy 20(3), 147 (2018)
3. Cena, A., Pistone, G.: Exponential statistical manifold. Ann. Inst. Statist. Math. 59(1), 27–56
(2007). https://doi.org/10.1007/s10463-006-0096-y
4. J. Newton, N.: 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
5. Kaniadakis, G.: Non-linear kinetics underlying generalized statistics. Physica A: Stat. Mech.
Appl. 296(3), 405–425 (2001). https://doi.org/10.1016/S0378-4371(01)00184-4
6. Krasnoseli’skii, M.A., Rutickii, J.B.: Convex functions and Orlicz spaces. Translated from the
first Russian edition by Leo F. Boron, P. Noordhoff Ltd., Groningen (1961)
7. 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
8. Matsuzoe, H., Scarfone, A.M., Wada, T.: Normalization problems for deformed exponential
families. In: Nielsen, F., Barbaresco, F. (eds.) Geometric Science of Information, pp. 279–287.
Springer International Publishing, Cham (2019)
9. Matsuzoe, H., Wada, T.: Deformed algebras and generalizations of independence on deformed
exponential families. Entropy 17(8), 5729–5751 (2015). https://doi.org/10.3390/e17085729
10. Montrucchio, L., Pistone, G.: A Class of Non-parametric Deformed Exponential Statistical
Models, pp. 15–35. Springer International Publishing, Cham (2019)
35
2.4 Conclusions
In this chapter, we investigated the behavior of the normalizing function near the
boundary of the domain of the parametrization. When Condition 2.2 occurs it was
proven that given any u belonging to the Musielak–Orlicz class, we have that ψ(αu)
converges when α tends to 1 and when u does not belong to the Musielak–Orlicz
class, it follows that ψ(αu) diverges when alpha tends to 1.
Assuming that Condition 2.2 does not occur, we proved that there exists a function
u not belonging the Musielak–Orlicz class such that ψ(αu) converges when α tends
to 1. Moreover, we observe that regardless the occurence of Condition 2.2, whenever
u is in the Musielak–Orlicz class, ψ(αu) converges since α tends to 1.
These results allow a more general class of normalizing functions and describe the
behaviour of it in the boundary domain and therefore make possible the characterization of a wider class of probability distributions for several conditions associated
to the convergence of the normalizing function.
Acknowledgements This work was supported by Coordenação de Aperfeiçoamento de Pessoal de
Nível Superior - Brasil (CAPES) - Finance Code 001 and Conselho Nacional de Desenvolvimento
Científico e Tecnológico (CNPq) (Procs. 309472/2017-2 and 408609/2016-8).
Disclaimer: Views and opinions expressed are those of the authors and do not necessarily represent
official positions of their respective companies.
References
1. Andrade, L.H.F.d., Vigelis, R.F., Vieira, F.L.J., Cavalcante, C.C.: Normalization and φ -
function: Definition and Consequences, pp. 231–238. Springer International Publishing, Cham
(2017)
2. Andrade, L.H., Vieira, F.L., F.Vigelis, R., Cavalcante, C.C.: Mixture and exponential arcs on
generalized statistical manifold. Entropy 20(3), 147 (2018)
3. Cena, A., Pistone, G.: Exponential statistical manifold. Ann. Inst. Statist. Math. 59(1), 27–56
(2007). https://doi.org/10.1007/s10463-006-0096-y
4. J. Newton, N.: 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
5. Kaniadakis, G.: Non-linear kinetics underlying generalized statistics. Physica A: Stat. Mech.
Appl. 296(3), 405–425 (2001). https://doi.org/10.1016/S0378-4371(01)00184-4
6. Krasnoseli’skii, M.A., Rutickii, J.B.: Convex functions and Orlicz spaces. Translated from the
first Russian edition by Leo F. Boron, P. Noordhoff Ltd., Groningen (1961)
7. 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
8. Matsuzoe, H., Scarfone, A.M., Wada, T.: Normalization problems for deformed exponential
families. In: Nielsen, F., Barbaresco, F. (eds.) Geometric Science of Information, pp. 279–287.
Springer International Publishing, Cham (2019)
9. Matsuzoe, H., Wada, T.: Deformed algebras and generalizations of independence on deformed
exponential families. Entropy 17(8), 5729–5751 (2015). https://doi.org/10.3390/e17085729
10. Montrucchio, L., Pistone, G.: A Class of Non-parametric Deformed Exponential Statistical
Models, pp. 15–35. Springer International Publishing, Cham (2019)