20
L. H. F. de Andrade et al.
The function ln φ is concave, negative in (0, 1), positive on (1, +∞) and its inverse
is the function exp φ , the so called deformed exponential [13].
The authors in [27] provided a generalization of exponential families of probability
distributions E( p) [17, 18], based on a wider class of deformed exponential functions,
calling it ϕ-families of probability distributions F
ϕ
c and the construction of these
families is based on Musielak–Orlicz spaces [11].
Another generalization of exponential families of probability distributions, but in
an infinite-dimensional setting, was studied in [29]. In [26], the authors studied the
2 -condition and its consequences on ϕ-families of probability distributions. More
specifically, the behavior of the normalizing function near the boundary of its domain
was analyzed, considering that the Musielak–Orlicz function c does not satisfy the
2 -condition.
To generalize the exponential families of probability E( p), the authors in [27]
required the assumption that the deformed exponential function ϕ satisfies the condition that there exists a function u 0 : T → (0, ∞) such that
T ϕ(c + λu 0 )dμ < ∞,
for all λ > 0, for every function c : T → R, for which
T ϕ(c)dμ = 1, considering
that ϕ(·) is a probability density.
Going further on the understanding of the behavior of the ϕ-families, in [24]
the authors investigated the importance of the existing condition imposed on the
deformed exponential ϕ and its relevance to connect arcs in generalized statistical
manifolds. Later, [1, 23] considered the impact of the same existing condition on
the behavior of the normalizing function ψ near the boundary of the domain of
parametrization of the ϕ-family of probability distributions. Still in this direction of
the study of normalization functions, in [20] the authors found an example that has the
same form of a deformed exponential function which does not satisfy the boundary
condition. More recently, the authors in [8] also discussed some conditions for the
normalization aspects on deformed exponentials. These aspects are of paramount
importance on the understanding of a statistical family of distributions.
Our aim in this chapter is to analyze the behavior of the normalizing function
near the boundary of its domain, considering deformed exponential functions which
satisfy or do not satisfy the boundary condition. We study the cases in the purely
atomic and in the non-atomic scenarios and discuss which are the constraints that
need to be satisfied so the probability family may be properly defined.
This chapter is organized as follows. In Sect. 2.2 we revisit the deformed exponential functions and the construction of the ϕ-families of probability distributions and
how the 2 -condition in Musielak–Orlicz functions influences the boundary of the
domain of the parametrization. Section 2.3 is devoted to all the cases of the behavior
of the normalizing function. Finally, in Sect. 2.4 we state our conclusions.
L. H. F. de Andrade et al.
The function ln φ is concave, negative in (0, 1), positive on (1, +∞) and its inverse
is the function exp φ , the so called deformed exponential [13].
The authors in [27] provided a generalization of exponential families of probability
distributions E( p) [17, 18], based on a wider class of deformed exponential functions,
calling it ϕ-families of probability distributions F
ϕ
c and the construction of these
families is based on Musielak–Orlicz spaces [11].
Another generalization of exponential families of probability distributions, but in
an infinite-dimensional setting, was studied in [29]. In [26], the authors studied the
2 -condition and its consequences on ϕ-families of probability distributions. More
specifically, the behavior of the normalizing function near the boundary of its domain
was analyzed, considering that the Musielak–Orlicz function c does not satisfy the
2 -condition.
To generalize the exponential families of probability E( p), the authors in [27]
required the assumption that the deformed exponential function ϕ satisfies the condition that there exists a function u 0 : T → (0, ∞) such that
T ϕ(c + λu 0 )dμ < ∞,
for all λ > 0, for every function c : T → R, for which
T ϕ(c)dμ = 1, considering
that ϕ(·) is a probability density.
Going further on the understanding of the behavior of the ϕ-families, in [24]
the authors investigated the importance of the existing condition imposed on the
deformed exponential ϕ and its relevance to connect arcs in generalized statistical
manifolds. Later, [1, 23] considered the impact of the same existing condition on
the behavior of the normalizing function ψ near the boundary of the domain of
parametrization of the ϕ-family of probability distributions. Still in this direction of
the study of normalization functions, in [20] the authors found an example that has the
same form of a deformed exponential function which does not satisfy the boundary
condition. More recently, the authors in [8] also discussed some conditions for the
normalization aspects on deformed exponentials. These aspects are of paramount
importance on the understanding of a statistical family of distributions.
Our aim in this chapter is to analyze the behavior of the normalizing function
near the boundary of its domain, considering deformed exponential functions which
satisfy or do not satisfy the boundary condition. We study the cases in the purely
atomic and in the non-atomic scenarios and discuss which are the constraints that
need to be satisfied so the probability family may be properly defined.
This chapter is organized as follows. In Sect. 2.2 we revisit the deformed exponential functions and the construction of the ϕ-families of probability distributions and
how the 2 -condition in Musielak–Orlicz functions influences the boundary of the
domain of the parametrization. Section 2.3 is devoted to all the cases of the behavior
of the normalizing function. Finally, in Sect. 2.4 we state our conclusions.