Citation: Liu, B.; Ananda, M.M.A.
A Generalized Family of
Exponentiated Composite
Distributions. Mathematics 2022, 10,
1895. https://doi.org/10.3390/
math10111895
Academic Editors: Sławomir
Nowaczyk, Rita P. Ribeiro and
Grzegorz Nalepa
Received: 1 April 2022
Accepted: 29 May 2022
Published: 1 June 2022
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Article
A Generalized Family of Exponentiated Composite Distributions
Bowen Liu
†
and Malwane M. A. Ananda *
,†
Department of Mathematical Sciences, University of Nevada, Las Vegas, NV 89154, USA; bowen.liu@unlv.edu
* Correspondence: malwane.ananda@unlv.edu
† These authors contributed equally to this work.
Abstract:
In this paper, we propose a new family of distributions, by exponentiating the random
variables associated with the probability density functions of composite distributions. We also derive
some mathematical properties of this new family of distributions, including the moments and the
limited moments. Specifically, two special models in this family are discussed. Three real datasets
were chosen, to assess the performance of these two special exponentiated-composite models. When
fitting to these three datasets, these three special exponentiated-composite distributions demonstrate
significantly better performance, compared to the original composite distributions.
Keywords:
composite models; goodness-of-fit; IG distribution; Weibull distribution; Pareto
distribution; exponential distributions; exponentiated models
MSC: 62P05; 62E99
1. Introduction
When dealing with right-skewed data, with a hump-shaped frequency distribution,
analysts, generally, begin with fitting to a commonly used right-skewed parametric distri-
bution. However, such a procedure, sometimes, fails to fit the data in a satisfactory way.
For example, lognormal distribution is a natural choice, when fitting to right-skewed data.
However, it does not provide satisfactory performance, when the upper tail is fairly large,
while Pareto distribution is more suitable in such situations. If Pareto distribution is chosen
instead, the lower tail features of the data cannot be captured, since the density of Pareto
distribution is, monotonically, decreasing. Thus, the concept of composite distribution has
arisen in the literature. The composite distributions are constructed, by combining two
parametric distributions at a specific threshold.
Such a concept is widely used in different areas, such as modeling insurance claim
size data [
1
–
9
], predicting the risk measures in insurance data analysis [
4
–
7
], fitting sur-
vival time data [
10
], and modeling precipitation data [
11
]. Such a concept demonstrates
impressive performances, when the data is characterized with a very heavy upper tail,
in which common distributions, such as normal or exponential distributions, cannot cap-
ture all of the data features. Due to the simplicity and the applicability of the concept,
the researchers developed a considerable number of composite distributions, including
Lognormal-Pareto [1], Weibull-Pareto [12], Weibull-Inverse Weibull [10], and so on.
The composite distributions seems proper, when modeling the data with heavy tails.
For example, both the one-parameter Inverse Gamma- Pareto (IG-Pareto) model [
6
] and
the one-parameter exponential-Pareto (exp-Pareto) model [
13
] were suggested as possible
models, for insurance data modeling. However, they still cannot provide a satisfactory
performance, when fitting to well-known insurance datasets, such as the Danish fire
insurance dataset. Thus, it is necessary to improve the model. In order to improve the
one-parameter IG-Pareto model, Liu and Ananda [
8
] proposed an exponentiated IG-Pareto
model, by exponentiating the random variable associated with the probability density
Mathematics 2022, 10, 1895. https://doi.org/10.3390/math10111895 https://www.mdpi.com/journal/mathematics
