IHJPAS. 36 (3) 2023 

437 

This work is licensed under a Creative Commons Attribution 4.0 International License 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

*Corresponding Author: kehindebashiru@uniosun.edu.ng 

 

 

 

 

Abstract  

A new family of distribution named Double-Exponential-X family is proposed. The proposed 

family is generated from the double exponential distribution. The forms of the probability densities 

and hazard functions of two distinct subfamilies of the proposed family are examined and reported. 

General properties such as moment, survival, order statistics, probability weighted moments and 

quartile functions of the models are investigated. A sub family of the developed family of double 

–Exponential-X family of the distribution known as double-Exponential-Pareto distribution was 

used to fit a real life data on the use of antiretroviral drugs. Montecarlo simulation of Efficacy of 

antiretroviral drugs is conducted to evaluate the performance of the model. The models were tested 

using some models diagnostic tests and it was revealed that the proposed model was better than the 

ones proposed before it from the same family and also,  the stochastic dominance method was used 

to affirm the best antiretroviral drugs used in the study.  

 

Keywords: Double Exponential Distribution, Hazard Function, Molecular Simulation, Stochastic 

Dominance. 

 

 

doi.org/10.30526/36.3.3377 

Article history: Received 25 January 2023, Accepted 14 March 2023, Published in July 2023. 

 

 

 

 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: jih.uobaghdad.edu.iq 

 

Double-Exponential-X Family of Distributions: Properties and Applications 

     Kehinde Adekunle Bashiru*  
Department of Statistics, Osun State University, 

Osogbo, Nigeria  

       Taiwo Adetola Ojurongbe  

Department of Statistics, Osun State University, 

Osogbo, Nigeria 

 

 Lawal Sola  

Department of Statistics, Osun State College of 

Technology, Esa Oke, Nigeria 

     Nureni .Olawale Adeboye  

Department of Statistics, Osun State University, 

Osogbo, Nigeria 

 

      Habeeb Abiodun Afolabi  

Department of Statistics, Osun State University, 

Osogbo, Nigeria 

 

https://creativecommons.org/licenses/by/4.0/
mailto:%20kehindebashiru@uniosun.edu.ng
mailto:kehindebashiru@uniosun.edu.ng
mailto:taiwo.ojurongbe@uniosun.edu.ng
mailto:slawal@gmail.com
mailto:nureni.adeboye@uniosun.edu.ng
mailto:habeeb.afolabi@uniosun.edu.ng


IHJPAS. 36 (3) 2023 

438 
 

1. Introduction 

In the past few decades, a variety of classical or general distributions have been widely used to 

structure data in the fields of engineering, actuarial, environmental science, biology, demography, 

economy, insurance, and finance. Despite this, there continue to be problems in each of the 

aforementioned fields. Naturally, this may be attributed to the fact that, as a result of the growth 

of science and technology, the underlying distributions are no longer able to suit the data set. 

Because of this, it is necessary to create new kinds of extended distributions that can handle these 

fresh difficulties. As a result, numerous methods have been proposed and researched to develop or 

introduce new families of distribution by including a parameter or several parameters in the 

baseline random variable model while providing a great degree of flexibility in actual data 

modeling, including beta-G [1,2], Gamma-G type-1 [3], Mc-G [4], Log-Gamma-G Type-2  [5], 

Gamma-G type-3 [6], Weibull-X family of distributions [7], odd generalized exponential-G [8,9], 

type-1 half-logistic family [10], Kumaraswamy-Weibull-generated family [11], new Weibull- G 

family [12], generalized transmuted- G [13] 

 Taking 𝑣(𝑡) to be the probability density function (PDF) of a random variable, say, which belongs 

to m and n, that is T∈ [𝑚, 𝑛] for -∞ ≤ m <  n <∞ and also, let 𝑊[𝐹(𝑥, 𝜉)] be a function of 

cumulative distribution function (CDF) of a random variable, say X which depends on the 

parameter 𝜉 which satisfies the conditions: 

i 𝑊[𝐹(𝑥, 𝜉)] is bounded between m and n 

ii. 𝑊[𝐹(𝑥, 𝜉)] is differentiable and monotonically increasing 

iii. 𝑊[𝐹(𝑥, 𝜉)] →m as x→-∞ and 𝑊[𝐹(𝑥, 𝜉)]→n as x→∞  

 The CDF of the T-X family of distribution as recently used by [14] as  

𝐺(𝑥) = ∫ 𝑣(𝑡)𝑑𝑡
𝑤(𝐹(𝑥;𝜉))

𝑚

 , 𝑥 ∈ 𝑅                                         (1) 

And the PDF corresponding to (1) is given as: 𝑔(𝑥) = {
𝑑

𝑑𝑥
𝑤[𝐹(𝑥; 𝜉)]} 𝑉{𝑤[𝐹(𝑥: 𝜉)]} 𝑥 ∈ 𝑅 

Several new families of distributions have been postulated using the concept of T-X family as 

proposed  [14]. Different functions of 𝑊[𝐹(𝑋; ∈)] have been employed in the literature for the T-

X family which are revealed in Table1. 

This research aims to introduce a new family of continuous distributions, which is named as 

Double Exponential-X (“DE-X” for short) family. This family's two distinct sub-models are 

looked into and discussed. Both real life and simulated data are used to illustrate the essence of the 

new model in the area of applied science.  

This paper's remaining sections are divided as follows: In the section titled "Double-Exponential 

X family," we define the DE-X family of distributions, provide a mathematical derivation of some 

of the family's fundamental statistical properties, including the moment, survival function, order 

statistics, probability weighted moment (PWM), and quartile function, and provide a maximum 

likelihood estimation method for determining the model parameters. The DE-Pareto Distribution 

sub-model constructed from this family was fitted to real- life data of viral load (Copies/ml) of 

some patients receiving antiretroviral medication. In order to calculate the average binding affinity 



IHJPAS. 36 (3) 2023 

439 
 

of Abacavir and Neviranpine as they docked (using molecular docking), simulated data (from 

simulation of molecular dynamics) was also used.  

 

Table 1: A few T-X Family Members Currently in Existence 

𝑾[𝑭(𝒙; 𝝃)] Range of 𝑿 Members of T – X family 

𝐹(𝑥, 𝜉) (0,1) Beta – G [1] 

− log[𝐹(𝑥; 𝜉)] (0, ∞) Gamma-GType-22) [15] 

− log[1 − 𝐹(𝑥; 𝜉)] (0, ∞) Gamma-GType-1[3] 

𝐹(𝑥, 𝜉)

1 − 𝐹(𝑥, 𝜉)
 

(0, ∞) Gamma-GType-3 [6] 

− log[1 − 𝐹(𝑥, 𝜉)] (0, ∞) Exponentiated T – X  [14]  

log {
𝐹(𝑥, 𝜉)

1 − 𝐹(𝑥, 𝜉)
} 

(−∞, ∞) Logistics – G [16] 

log[− log{1 − 𝐹(𝑥, 𝜉)}] (0, ∞) The logistic – X family [8,9] 

 

2.  Materials and Methods 

2.0 The Double Exponential-X Family 

Given the PDF of the double exponential distribution 

𝑉(𝑥; 𝑏; µ) =
1

2𝑏
𝑒

|𝑥−µ|

𝑏  ( 21)                                                                              (2) 

And the corresponding CDF with high reliability on the support is given by: 

𝐹(𝑥; µ; 𝑏) =
1

2
𝑒

−(µ−𝑥)

𝑏       if𝑥 ≤ µ                                                                     (3) 

𝐹(𝑥; µ; 𝑏) = 1 −
1

2
𝑒

−(𝑥−µ)

𝑏       if𝑥 ≥ µ                                                              (4) 

Then, if 𝑥 ≥ 0, then the CDF changes to 

𝐹(𝑥; 𝑏) = 1 −
1

2
𝑒

−
𝑥

𝑏                                                                                       (5) 

And its corresponding PDF is given as: 

𝑉(𝑥; 𝑏) =
1

2𝑏
𝑒

−𝑥

𝑏                                                                                              (6) 

If 𝑥 follows (6) and setting 𝑊[𝐹(𝑥, 𝜉)]= 
1

1−𝐹(𝑥,𝜉)
 in (1), we define the CDF of the Double 

Exponential-X family by 

G(𝑥, 𝑏, 𝜉)  =  
1

2
[1 − 𝑒

−
1

𝑏
(𝑊[𝐹(𝑥,𝜉)])

]  =   
1

2
−

1

2
𝑒

−
1

𝑏
(𝑊[𝐹(𝑥,𝜉)])

,  b, 𝜉 > 0  ,   𝑥 ∈ 𝑅     (7) 

Where  the baseline distribution's CDF, which is dependent on the vector parameter, is denoted by 

𝐹(𝑥, 𝜉), 

 𝜉 and 𝑊[𝐹(𝑥, 𝜉)]= 
1

1−𝐹(𝑥,𝜉)
 is a special transformation. The corresponding PDF is obtained via 

the first derivative of the CDF, G(𝑥, 𝑏, 𝜉) and it is shown by:                                                                                                                                                       

𝑔(𝑥, 𝑏, 𝜉) =
1

2𝑏
[

𝑑

𝑑𝑥
(𝑊[𝐹(𝑥, 𝜉)])] 𝑒

−{
1
𝑏

(𝑊[𝐹(𝑥,𝜉)])}
,     𝑏, 𝜉 > 0           (8) 

𝑔(𝑥, 𝑏, 𝜉)= 𝑔(𝑥, 𝑏, 𝜉) =
1

2𝑏
[(𝑊[𝐹(𝑥, 𝜉)])′]𝑒

−{
1

𝑏
(𝑊[𝐹(𝑥,𝜉)])}

,   where  (𝑊[𝐹(𝑥, 𝜉)])′ =

𝑑

𝑑𝑥
(𝑊[𝐹(𝑥, 𝜉)]) and 𝑏, 𝜉 > 0 



IHJPAS. 36 (3) 2023 

440 
 

The motivations behind using DE-X family in the real life situations include: 

  It is a popular technique for inserting extra parameter(s) to produce an expanded baseline 
distribution, 

  To enhance the properties of the traditional distributions, 

 To continuously offer better fits compared to other produced distributions with the same 
or more parameters, 

2.1  Special Sub-Models 

Most often for one reason or another, extended versions of distributions are introduced for the 

following reasons; 

1. An alternate model to the existing distribution that has been successfully applied in the past 

2. A flexible statistical model that provides a good match to the data 

3. A model with closed CDF, Survival Function (SF), Hazard Rate Function( HRF) forms and a 

strong propensity to lessen estimation challenges. 

Thus, we introduce two unique sub-models of the DE-X family that have at least one of the 

aforementioned characteristics. 

2.1.1  The DE-Pareto (DE-P) distribution 

The PDF and the CDF of the Pareto random variable are 

 f(𝑥, ⍬, 𝛼) =
𝛼(𝛼⍬)

𝑥⍬+1
 and F(𝑥, 𝛼, 𝜃) = 1 − (

𝛼

𝑥
)

⍬

, 𝑥>0, 𝛼, 𝜃>0, . 

 Then the CDF and the PDF of the DE-P model are given by (9) and (11) respectively. 

𝐺(𝑥, 𝑏, 𝛼, 𝜃) =
1

2
−

1

2
𝑒

−
1

𝑏
(

𝑥

𝛼
)

𝜃

, b, 𝜃, 𝛼>0                                                                  (9) 

Hence, 𝐺(𝑥, 𝑏, 𝛼, 𝜃) =
1

2
(1 − 𝑒

−
1

𝑏
(

𝑥

𝛼
)

𝜃

)b, 𝜃, 𝛼 > 0(10)                                         (10) 

 

Hence,𝑔(𝑥, 𝑏, 𝛼, 𝜃)=
⍬

2𝛼𝑏
(

𝑥

𝛼
)

⍬−1

𝑒
−

1

𝑏
(

𝑥

𝛼
)

⍬

, 𝑥, b,𝜃, α > 0                                         (11) 

 

Plots of the DE-P density and Hazard rate function for selected parameter values are presented in 

Fig.1 and Fig.2 respectively below: 

 
Figure.1:PDF plot at fixed b =1 and increasing 𝛼 𝑎𝑛𝑑 𝜃 



IHJPAS. 36 (3) 2023 

441 
 

 
Figure.2:Hazard plot at fixed b =1 and increasing 𝛼 𝑎𝑛𝑑 𝜃 

 

2.1.2 The DE-ParetoII (DE-P2) 

Given the PDF and CDF of the Pareto type-2 distribution𝑓(𝑥, 𝑎, 𝑏) =
𝑏

𝑎
(

𝑎

𝑥−𝑎
)

𝑏+1

, b,a, 𝑥 > 0   and 

𝐹(𝑥; 𝑎, 𝑏) = 1 − (
𝑎

𝑥−𝑎
)

𝑏

, x, b,a,> 0.  

Then the CDF and PDF of DE-PII model are given as 

𝐺(𝑥, 𝑏, 𝑎, 𝜉) =  
1

2
[1 − 𝑒

−
1

𝑏
(

𝑥−𝑎

𝑎
)

𝜃

], a,b,𝜃 > 0;  𝑥 > 0 0                                (12) 

For the PDF,  

𝑔(𝑥, a, b, 𝜉) =  
𝜃

2𝑏𝑎
(

𝑥−𝑎

𝑎
)

𝜃−1

𝑒
−

1

𝑏
(

𝑥−𝑎

𝑎
)

𝜃

, a, b, 𝜃 > 0  ; 𝑥> 0                         (13) 

Fig.3 and Fig.4 give the plots of the PDF and HRF of DE-PII distribution for selected parameter 

values. 

 
Figure.3 The pdf of double exponential-ParetoII distribution at different values of parameters 

 



IHJPAS. 36 (3) 2023 

442 
 

 
Figure.4 showing the hazard of double exponential-ParetoII distribution at different values of parameters 

 

2.2  Basic Mathematical Properties of DE-Pareto 

Here, we provide the basic properties of DE-X family of distributions.          

2.2.1 Moments of DE-Pareto 

Assuming  X follows DE-X family with density 𝑔(𝑥, a, b, 𝜉), then its 𝑟𝑡ℎ moment is  

µr
1=∫ xr

∞

−∞
 g(x, a, b, ξ)dx 

E(Xr)  = ∫ xr
∞

0

⍬

2αb
(

x

α
)

⍬−1

e
−

1

b
(

x

α
)

⍬

dx                                                                      (14) 

E(Xr)  =   
1

2
(α √b

θ
)

r
┌ (

r

θ
+ 1) 

E(X)=   
α √b

θ

2
  ┌ (

1

θ
+ 1);  r =1                                                                         (15) 

Variance: 

Var(X)=  (α √b
θ

)
2

[
1

2
┌ (

2

θ
+ 1) − (

1

2
┌ (

1

θ
+ 1))

2

]                                                                  (16) 

2.2.2 Survival Function of DE-Pareto 

The survival function of DE-X family is given as 

𝑆(𝑥) = 1 − 𝐺(𝑥, 𝑏, 𝛼, 𝜃)                                                 (17) 

𝑆(𝑥)  =  
1

2
(1 + 𝑒

−
1

𝑏
(

𝑥

𝛼
)

𝜃

)                                                                                               (18) 

2.2.3 Probability Weighted Moments of DE-Pareto 

The Probability Weighted Moment (PWM) of DE-X random variable is 

𝑀𝑟𝑠(𝑥) = ∫ 𝑥
𝑠∞

−∞
𝑔(𝑥, 𝑏, 𝑎, 𝜉)𝐺(𝑥, 𝑏, 𝑎, 𝜉))𝑠𝑑𝑥                                                                               (19) 

2.2.4 Hazard Function of DE-Pareto 

𝐻(𝑥; 𝛼, 𝑏, 𝜃)  =  
𝑔(𝑥)

𝑆(𝑥)
 =   

⍬

2𝛼𝑏
(

𝑥

𝛼
)

⍬−1
𝑒

−
1
𝑏

(
𝑥
𝛼

)
⍬

1

2
(1+𝑒

−
1
𝑏

(
𝑥
𝛼

)
𝜃

)

 =  
⍬

𝛼𝑏
(

𝑥

𝛼
)

⍬−1
𝑒

−
1
𝑏

(
𝑥
𝛼

)
⍬

1+𝑒
−

1
𝑏

(
𝑥
𝛼

)
𝜃                             (20) 

2.2.5 Renyi Entropy of DE-Pareto 

𝛿𝑟  =   
1

1−𝑟
𝑙𝑜𝑔 ∫ 𝑓 𝑟 (𝑥)𝑑𝑥

∞

0
  =  

1

1−𝑟
 𝑙𝑜𝑔 ∫ (

⍬

2𝛼𝑏
(

𝑥

𝛼
)

⍬−1

𝑒
−

1

𝑏
(

𝑥

𝛼
)

⍬

)

𝒓

𝑑𝑥
∞

𝟎
                              (21) 



IHJPAS. 36 (3) 2023 

443 
 

𝛿𝑟 =   
1

1−𝑟
 𝑙𝑜𝑔 (

𝑏𝜃

2𝑟𝛼𝑏
√

𝑟

𝑏

𝜃
)

𝑟

(
𝛼

𝜃
√

𝑏

𝑟

𝜃
)  ┌ (𝑟 −

𝑟

𝜃
+

1

𝜃
)                                                                    (22) 

2.3. Estimation 

We describe the maximum likelihood method of estimation in this section for the unknown 

parameters of the DE-X family. The method of Maximum Likelihood estimation is adjudged to be 

the best method of estimating parameters of distributions. 

2.3.1 Maximum Likelihood Estimation DE-Pareto 

In this, let 𝑋1, 𝑋2, … , 𝑋𝑘 be the observed values for DE-X distribution with parameters𝑏, 𝑎, 𝜉. The 

total log-likelihood function corresponding to (11) is presented as Log 

 L(𝑥, 𝑏, 𝑎, 𝜉) =  ∏
⍬

2𝛼𝑏
(

𝑥

𝛼
)

⍬−1

𝑒
−

1

𝑏
(

𝑥

𝛼
)

⍬
𝑛
𝑖=1                                                                                (23) 

 

3.  Results  

3.1   Fitting the Model to Molecular Dynamics Simulated Data 

The two approved HIV drugs were obtained from The Drug bank archive online, modeled using 

(Spartan’14 version 1.18 by wave function Inc) 

i. Geometry optimization and calculation of the two drugs were done using density functional 

theory (DFT) with 6-31G* basis set to obtain the molecular parameters responsible for the 

cytotoxicity of the compounds under investigation and then saved as pdb files for simulation study  

ii. The protein receptors responsible for HIV(that is, the HIV-causing organisms) used were gotten 

from protein data bank (PDB)   [18]  

iii. with no missing residues and a single strand were obtained by deleting multiple ligands, non-

protein and other protein parts from the pdb files. Autoduck tools version 1.5.6 was used to locate 

the binding site and to add hydrogen to the prepared protein receptor and finally converted to a 

protein data bank format (pdbqt) for the molecular simulation. Autodockvina were used for the 

simulation process and the simulation was repeated for seven different in order to obtain the most 

stable conformation of the drug-receptor complex and post docking analysis was done using 

EduPymolversion 1.7.4.4 and discovery studio 4.1 visualizer. 

 

 

 
Figure 5a: Fitting the Models to Neviranpine data (Obtained via Molecular Simulation) 



IHJPAS. 36 (3) 2023 

444 
 

From the figure 5a above, we can say that the selected distributions approximately follow the 

distribution of the data, thus, making the distributions suitable for data collected when using 

Neviranpine to dock HIV protein receptor 

 

 
Figure 5b: Fitting the Models to Abacavir data (Obtained via Molecular Simulation) 

From the Fig. 5b above, we can say that the selected distributions approximately follow the distribution of the data, 

thus, making the distributions suitable for data collected when using Abacavir to dock HIV protein receptor 

 

Table1. Fitting cells table with the including numbers 

Model  Parameter  Criteria Mean Affinity 

EPD  �̂�
= 0.029978(0.034) 
�̂� = 0.9980(0.367) 

�̂� = 0.996486(0.648) 

AIC=303.405 

BIC=303.6433 

CAIC=309.405 

1.663157(0.902308) 

DEPD  �̂�=1.978e+06(1.639e-
02) 

 

�̂� = 1.503e-
04(5.307e-05) 

 

�̂�=9.024e-01(2.211e-
04) 

 

AIC =-27203950 

BIC= -27203950 

CAIC=27203944 

 

0.4511979(0.225) 

PD  �̂�= 8.089e+03 
(7.567e-02) 

�̂� =(3.372e+03)  
9.268e-02 

AIC=-810083 

BIC=-810082.8 

CAIC=-810080.6 

3.3724(0.41697) 

 

From table 2 above, we can say that the parameter of the model is stable and the Double 

exponential Pareto distribution (DEPD) appears to be the best of all the competing model (model 

with the lowest information criteria is the best of all the models). The 45.1198% of the binding 

affinity is attributed to Neviranpine.  



IHJPAS. 36 (3) 2023 

445 
 

 
Fig 4: Investigating Stochastic Dominance 

From fig 4 above, we can say that Neviranpine is first order (Stochastically) dominated Abacavir. 

That is, the mean effect of Nevirapine has dominated that of Abacavir when it comes to the viral 

load measurement (copies/ml) from the patients subjected to Anti-retroviral therapy. 

 

 

 

 

 

 

 

 

        

3D Structures of Nevirapine                         3D of Abacavir 

Figure 5: The Ligands (Nevirapine and Abacavir) used as the drug in Anti-retroveral therapy 

 

 



IHJPAS. 36 (3) 2023 

446 
 

 
3D Receptor Responsible for HIV (Protein responsible for HIV) 

 

 
Interaction of Abacavir and 3PLT 

 
Interaction of Nevirapine and 3LPT 

Figure 6. The Docking Result of Abacavir and Nevirapine 

 

The Abacavir and Nevirapine were used as the Ligands to dock HIV protein receptor, the fig 6 

showed interaction between the ligands and HIV protein receptor.



 IHJPAS. 36 (3) 2023 
 

 

447 
 

 

 

Table 5: The Molecular Properties of the Drug used in this study (Molecular Docking) 

Descriptors Nevirapine Abacavir 

Molecular Formula C15H14N4O C14H18N6O 

Molecular Weight 266.30amu 286.34amu 

Area 281.25A2 305.46A2 

Volume 269.04A3 285.66A3 

Polarizability 62.14 63.34 

Hydrogen Bond Donor (HBD) 1 2 

Hydrogen Bond Acceptor (HBA) 5 5 

Polar Surface Area  (PSA) 36.84 67.47 

Ovality 1.40 1.46 

Log P 0.13 0.16 

Dipole Movement 2.33 1.93D 

HOMO -5.80Ev -5.28eV 

LUMO -1.32eV -0.16eV 

BG 4.58eV 5.12eV 

 

4.  Discussion 

From Table5, the properties of the drugs used in determining the toxicity are clearly displayed, as 

behavior, and the general effect of each drug on the body system.  Parameters such as HBA, HBD, 

PSA, LogP and polarizability are used to determine if a particular compound is a good drug 

candidate. This reveals the active properties of the drugs used in determining the cytotoxicity of 

each drug towards the receptor responsible for HIV. Failure of most drugs in clinical experiments 

is a result of there poor ADME (Absorption, Distribution, Metabolism and Excretion) properties, 

hence for a ligand to be chosen as a drug candidate with good ADME properties HBD must be less 

than five, HBA must be less than ten and its molecular weight in g/mol must be below 500, any 

ligand that violates this set of rules might result to the problem if ingested [19].  

Polar Surface Area (PSA) gives information about the ability of a drug-like compound to penetrate 

into the cell membrane. It has been established that ligands with high PSA value greater than 140Å2 

will have poor penetrating power into the cell membrane. On the other hand, molecules with PSA 

value below 60Å2 will have high penetrating power [20]. Hence, we are expecting high penetrating 

power into the cell membrane from Abacavir and Nevirapine, and these are used to determine if a 

particular compound is good drug candidate.  

The docking result used for stochastic dominance in order to determine the most effective drugs 

that perform actively in curing the diseases shows better binding affinity and inhibition constant 

from Nevirapine as compared to Abacavir and this is an indication that Nevirapine drug is more 

potent for curing the disease(HIV).  

 

5. Conclusion 

A new family of distribution (Double Exponential-X family) has been introduced. Two distinct 

sub-models of the distribution were discussed. It is shown that the two special models' densities 

can generally be skewed to the left or right, symmetrical, reverse j-shaped, and can also have both 

growing and decreasing, as well as, of course, unimodal. The maximum likelihood estimation 

approach is used to estimate the parameters. A molecular simulation study that compares the newly 



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built model's performance to previous well-known distributions is presented. There are derived 

and explained several fundamental mathematical characteristics of the new family. Finally, real-

life applications of the model to data sets on the effectiveness of various antiretroviral medications 

were provided, and they compare favorably to other well-known distributions in terms of data fit, 

and the efficacy of the pharmaceuticals utilized was compared using the stochastic dominance 

approach. 

 

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