Short Communications
On the $q$-continuous natural exponential family
B. S. Nahlaab a Laboratory of Probability and Statistics, University of Sfax, Tunisia
b Higher Institute of Environmental Sciences and Technologies, University of Carthage, Tunisia
Abstract:
In this paper, we present the concept of
$q$-natural exponential families
within the framework of
$q$-calculus, which extends the classical notion by
utilizing the
$q$-kernel
$e_q^{\theta x f(x)^{q-1}}$ in place of the
traditional exponential kernel
$e^{\theta x}$. This approach allows us to
rederive several established results and explore additional parallels with
exponential families. The kernel function has been examined in various
contexts to evaluate specific integral forms. Inspired by applications in
statistical mechanics, we investigate the properties of the
$q$-natural
exponential family (
$q$-NEF). Additionally, we derive the
$q$-Laplace
transform of a density function and demonstrate the convexity of this
function. We establish that the
$q$-NEF is uniquely characterized by its
$q$-variance function within its
$q$-means domain, which is often simpler
than the corresponding generating probability measure. This characterization
is facilitated by the relationship between the
$q$-mean and the
$q$-variance
function of a
$q$-NEF. Moreover, we provide explicit calculations for the
$q$-NEFs and
$q$-variance functions of the
$q$-Gaussian and
$q$-exponential
distributions.
Keywords:
\hskip-1pt $q$-Laplace transform, $q$-exponential distribution, $q$-Gaussian distribution, $q$-variance. Received: 25.02.2025
Revised: 30.03.2025
DOI:
10.4213/tvp5795
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