The slash distribution is an extended version of the normal distribution. It is characterized by the ratio of two separate random variables: one following a normal distribution and the other following a power of the uniform distribution. Therefore, we define a slash distribution for variable
S as:
where
,
is independent of
and
; its representation can be seen in Johnson et al. [
1]. The distribution in question exhibits heavier tails compared to the normal distribution, indicating a higher level of kurtosis. The characteristics of this particular distribution are explored in detail in the works of Rogers and Tukey [
2] and Mosteller and Tukey [
3]. Kafadar [
4] delves into the topic of maximum likelihood estimation for the location and scale parameters. Wang and Genton [
5] present a multivariate version of the slash distribution as well as a multivariate skew version. The slash distribution is further extended by Gomez and Venegas [
6] through the incorporation of the multivariate elliptic distributions. This methodology to increase the weight of the queues has also been used in distributions with positive support. To name a few, we mention the works of Olmos et al. [
7] in the half-normal and Rivera et al. [
8] in the Rayleigh model, among others. Based on the work of Rivera et al. [
8], the scale mixture of Rayleigh (SMR) model is proposed. We say that
with
and
if the probability density function (pdf) of
Y is
Also, a necessary distribution in the development of this paper is the gamma distribution, whose pdf is given by
where
. Its corresponding cumulative distribution function (cdf) is denoted by:
Shanker [
9] introduced the Akash distribution and applied it to real lifetime data sets from medical science and engineering. Thus, we say that a random variable (r.v.)
Y has an Akash model (AK) with shape parameter
if its pdf is
where
and we denote it by
The parameter
is a shape parameter, and if we add a scale parameter the pdf is given by
where
is a scale parameter and
is a shape parameter. We denote it by
Extensions of the AK distribution are carried out by Shanker and Shukla [
10,
11], among others. Both extensions consider adding a parameter and we will compare them with the new distribution. The two-parameter Akash distribution (TPAD) introduced by Shanker and Shukla [
10] has the following pdf:
where
and we denote it by
The power Akash distribution (PAD), introduced by Shanker and Shukla [
11], has the following pdf:
where
and we denote it by
The main motivation of this work is to introduce an extended version of the AK distribution given in Equation (
6), making use of the slash methodology, in order to obtain a new distribution with greater kurtosis to be able to accommodate outliers. Pronounced fluctuations in the data sets encountered in such diverse disciplines as economic and actuarial sciences, environmental and earth sciences, among others, are very frequent. Thus, heavy-tailed models are necessary to perform better modelling in the presence of extreme values. For example, the normal distribution does not perform well in modelling data sets with extreme observations. We must therefore resort to heavy-tailed distributions. For example, in problems in which the involved r.v. has a high kurtosis, the probability that a rare event occurs can be highly underestimated if a model without heavy tails is used, which is solved by using a model with these characteristics. In the economy, practical examples of rare events are pandemics, and the 2008–2009 financial crisis, to name a few. In geology, a rare event might be a mega earthquake or a sudden eruption of a volcano that has been dormant for centuries.
The paper is structured as follows: in
Section 2 we deliver our proposal and present its properties. In
Section 3, we perform inference using the method of moments and maximum likelihood via the EM algorithm and a simulation study is also carried out. In
Section 4, we apply the distribution to a real data set and compare it with other extensions of the AK distribution. Finally, in
Section 5, we provide the main conclusions.