Chapter 2
On Normalization Functions and
ϕ-Families of Probability Distributions
Luiza H. F. de Andrade, Francisca L. J. Vieira, and Charles C. Cavalcante
Abstract ϕ-families of probability distributions are a general representation to
deformed exponential models which brings interesting insight on the geometrical
aspects of the distribution family. In the ϕ-family, the analogue of the cumulantgenerating function is a normalizing function. This function plays an important role
in the statistical model and we investigate the behaviour of the function near the
boundary of the domain of the parametrization in order to provide the precise conditions of validity of the ϕ- family model. We discuss the conditions for existence of
the statistical model when considering ϕ-functions and the required conditions for
the normalization function so we can provide a complete understanding about the
investigated family of probability distribution.
2.1 Introduction
Deformed exponentials functions were initially studied in [13] and refer to the generalization of the classical exponential function by a given function φ of [0, ∞), which
is strictly positive on [0, ∞). The function φ is used to define the ln φ (φ-logarithm)
as:
ln φ (u) =
u
1
1
φ(x)
dx, u > 0.
L. H. F. de Andrade
Department of Natural Sciences, Mathematics and Statistical,
Federal Rural University of Semi-Arid Region, Mossoró, RN, Brazil
e-mail: luizafelix@ufersa.edu.br
F. L. J. Vieira
Department of Mathematics, Regional University of Cariri, Juazeiro do Norte-CE, Brazil
e-mail: leidmar.vieira@urca.br
C. C. Cavalcante (B)
Department of Teleinformatics Engineering, Federal University of Ceará, Fortaleza, CE, Brazil
e-mail: charles@ufc.br
© Springer Nature Switzerland AG 2021
F. Nielsen (ed.), Progress in Information Geometry,
Signals and Communication Technology,
https://doi.org/10.1007/978-3-030-65459-7_2
19
On Normalization Functions and
ϕ-Families of Probability Distributions
Luiza H. F. de Andrade, Francisca L. J. Vieira, and Charles C. Cavalcante
Abstract ϕ-families of probability distributions are a general representation to
deformed exponential models which brings interesting insight on the geometrical
aspects of the distribution family. In the ϕ-family, the analogue of the cumulantgenerating function is a normalizing function. This function plays an important role
in the statistical model and we investigate the behaviour of the function near the
boundary of the domain of the parametrization in order to provide the precise conditions of validity of the ϕ- family model. We discuss the conditions for existence of
the statistical model when considering ϕ-functions and the required conditions for
the normalization function so we can provide a complete understanding about the
investigated family of probability distribution.
2.1 Introduction
Deformed exponentials functions were initially studied in [13] and refer to the generalization of the classical exponential function by a given function φ of [0, ∞), which
is strictly positive on [0, ∞). The function φ is used to define the ln φ (φ-logarithm)
as:
ln φ (u) =
u
1
1
φ(x)
dx, u > 0.
L. H. F. de Andrade
Department of Natural Sciences, Mathematics and Statistical,
Federal Rural University of Semi-Arid Region, Mossoró, RN, Brazil
e-mail: luizafelix@ufersa.edu.br
F. L. J. Vieira
Department of Mathematics, Regional University of Cariri, Juazeiro do Norte-CE, Brazil
e-mail: leidmar.vieira@urca.br
C. C. Cavalcante (B)
Department of Teleinformatics Engineering, Federal University of Ceará, Fortaleza, CE, Brazil
e-mail: charles@ufc.br
© Springer Nature Switzerland AG 2021
F. Nielsen (ed.), Progress in Information Geometry,
Signals and Communication Technology,
https://doi.org/10.1007/978-3-030-65459-7_2
19