363 

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

 IHJPAS.37 (2) 2024 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: jih.uobaghdad.edu.iq 

PISSN: 1609-4042, EISSN: 2521-3407 

 

 

    Khalaf  H. Al-Habib 1* , Mundher A. Khaleel 2   and Hazem Al-Mofleh3   

1,2 Department of Mathematics, Collage of Computer Sciences and Mathematics, Tikrit University, Tikrit, Iraq. 
3 Department of mathematics, Tafila Technical University, Jordan. 

*Corresponding Author. 
 

 

 

Abstract  

There is a desperate need for extended versions of the classical distributions. There have been 

attempts to find novel families of probability distributions that widen existing families and provide 

great flexibility in data modeling in a number of application areas, including lifetime analysis, 

finance, and insurance. In this paper, we introduce a new family of distributions based on [0,1] 

Truncated and propose a new extension for the exponential distribution. The new distribution is 

called Truncated Nadarajah-Haghighi Distribution, symbolized with {[0,1]TNHE}. This study 

aims to derive some statistical properties for the new distribution, such as the quantile function, 

the mixture representation for the probability density function, the moments, the incomplete 

moments, the stress strength, the Rényi entropy, and the Shannon entropy. In addition, we 

estimated the parameters using the maximum likelihood method and proposed the simulation and 

application of the selected parameters using the statistical software R. 

Keywords: exponential distribution, entropy, MLE, moments, Nadarajah-Haghighi. 

 

1. Introduction 

Statistical distributions are important part of our lives. They enable us to understand the world 

around us and make informed decisions. They also help us to recognize trends and opportunities. 

In recent years, the modeling of lifetime data has become an important research topic. Numerous 

studies have been published on this topic with the aim of introducing new statistical methods for 

dealing with lifetime phenomena. Several families of statistical distributions have been used in the 

last decades in a variety of fields, such as engineering, economics, medicine, demography, etc. In 

this paper, we propose a new extension of the exponential distribution based on the family of [0,1] 

truncated Nadarajah-Haghighi G distributions. The family of truncated Nadarajah-Haghighi-G 

distributions is proposed. The generated families generalize and extend most of the formal 

distributions. Some of the generators are Beta-G [1] and Exponential-G [2]. The Weibull-G family 

was proposed by [3] and the generalized transmuted-G was studied and introduced by [4]. The 

Gompertez-G family  from [5]. the generalized odd Lindley-G family was proposed by [6], while 

the generalized odd gamma-G family was introduced by [7], and the Marshal-Olkin alpha-power 

family was proposed in [8]. The Gamma-Kumaraswamy G- family of distributions was introduced 

Received 19 March 2023               Accepted 14 May 2023                       Published 20 April 2024 

 

 

Statistical Propoerties and Application for [0,1] Truncated Nadarajah-

Haghighi Exponential Distribution 
 

doi.org/10.30526/37.2.3349 

https://creativecommons.org/licenses/by/4.0/
https://jih.uobaghdad.edu.iq/index.php/j/index#1609-4042
https://jih.uobaghdad.edu.iq/index.php/j/index#2521-3407
https://orcid.org/0009-0005-1045-2777
mailto:khalaf.h.khalaf35385@st.tu.edu.iq
https://orcid.org/0000-0001-8827-3748
mailto:mun880088@tu.edu.iq
https://orcid.org/0000-0003-3430-2464
mailto:Almof1hm@cmich.edu


 IHJPAS. 37 (2) 2024 

 364 
 

by [9]. The Marshall-Olkin Topp Leone-G family was proposed by [10]. The Marshal-Olkin-

Weibull-H family was introduced by [11]. The Odd-Chen family was introduced by [12]. 

Researchers have derived a truncated distribution from a parent distribution, such as a normal or 

exponential distribution, by bounding the random variable from below, from above, or from both. 

[13] The authors discussed the [0, 1] truncated inverted inverted gamma distribution as a special 

case of the CDF, moments, mean, variance, skewness, kurtosis, median, and characteristic 

function. Following the same method, [14] introduced the [0,1]-truncated inverse Weibull family. 

More recently, [15] introduced and studied a new distribution called [0, 1]-truncated exponential 

Gompertz distribution. In addition, many researchers have introduced new extensions to the 

exponential distribution, such as Gupta et. al. (2001) [16], Generalized exponential distribution. 

[17], a new generalization of the exponential Pareto distribution [18], the Weibull exponential 

distribution [19] and the beta exponential distribution. [20]. We use the modified exponential 

distribution ([0,1]) centered on the Nadarajah-Haghighi distribution to generate a new family of 

distributions and achieve greater flexibility than the existing submodels. 

The article is organized as follows: Section 2 presents a useful [0, 1] truncated K-MD. Section 3 

introduces the [0, 1] Truncated Nadarajah-Haghighi-M family of distributions. In addition, Section 

4 contains the statistical properties of the [0, 1] TNHE distribution. To estimate the parameters of 

this new distribution, the MLE method is used, which is presented in Section 5. In Section 6, the 

estimates are validated by the simulation process of the TNHE distribution. In Section 7, a real 

data set is used to illustrate the effectiveness of the TNHE distribution. The conclusions are 

presented in Section 8. 

 

2. [0,1] Truncated K-MD 

In this paper, we have generated a new family of continues distributions based on [0,1] truncated 

CDF K-M, named [0,1] K-M as noted by [0,1] TK-M distributions.  

Suppose that 𝑀(𝑥), 𝑚(𝑥) is any continuous CDF and PDF, respectively of the random variable 

𝑋, and assume that 𝐾(. ), 𝑘(. ), respectively represent the CDF and PDF of any continuous 

distribution on the interval [0, ∞). The suggested general formula of CDF for this class depends 

on the synthesis of K with M is 

𝐹(𝑥)𝑇𝐾−𝑀 =  
𝐾[𝑀(𝑥)] − 𝐾(0)

𝐾(1) − 𝐾(0)
 

(1) 

Now, let  𝐾(0) = 0, then the CDF in (1) can be written as: 

𝐹(𝑥)𝑇𝐾−𝑀 =  
𝐾[𝑀(𝑥)]

𝐾(1)
 

(2) 

And its associated PDF will be  

𝑓(𝑥)𝑇𝐾−𝑀 =
𝑘[𝑀(𝑥)]𝑚(𝑥)

𝐾(1)
 

(3) 

  

3. [0,1] Truncated Nadarajah-Haghighi-M family  

Here we will propose a new family of [0,1] Truncated based on Nadarajh-Haghighi distribution. 

The Nadarajh-Haghighi distribution was introduced by [21] as an  extention of the exponential 

distribution.  The N-H distribution  has the CDF and PDF, as folllows: 

   𝐾(𝑋) = 1 − 𝑒1−[1+𝑏𝑥]𝑎
                                          𝑥 > 0,      𝑎 , 𝑏 > 0       (4) 

 

   𝑘(𝑥) = 𝑎𝑏[1 + 𝑏𝑥]𝑎−1 𝑒1−[1+𝑏𝑥]𝑎
                  𝑥 > 0,      𝑎 , 𝑏 > 0   (5) 

 



 IHJPAS. 37 (2) 2024 

 365 
 

Then, 𝐾[𝑀(𝑥)] =  1 −  𝑒1−[1+𝑏𝑀(𝑥)]𝑎
, 𝑘[𝑀(𝑥)] =  𝑎𝑏[1 + 𝑏𝑥]𝑎−1 𝑒1−[1+𝑏𝑀(𝑥)]𝑎

 

So that equations (2), (3) can be rewritten as follows: 

    𝐹(𝑥) =  
1− 𝑒1−[1+𝑏𝑀(𝑥,𝜑)]𝑎

1− 𝑒1−[1+𝑏]𝑎  

 

(6) 

     𝑓(𝑥) =
𝑎𝑏[1+𝑏𝑀(𝑥,𝜑)]𝑎−1 𝑒1−[1+𝑏𝑀(𝑥,𝜑)]𝑎

 𝑚(𝑥,𝜑)

1− 𝑒1−[1+𝑏]𝑎  
(7) 

The equations (6), (7), respactively represent the CDF, PDF of [0,1] Truncated Nadarajah-

Haghighi-M family of distributions, where 𝑀(𝑥, 𝜑), 𝑚(𝑥, 𝜑) are the CDF and PDF of the baseline 

distribution with vector of parameters 𝜑. 

The exponential distribution was introduced by the random variable 𝑋 provided that the following 

CDF and PDF: 

 

 

 𝑀(𝑥, 𝜆) = 1 − 𝑒−𝜆𝑥 

 𝑚(𝑥, 𝜆) = 𝜆𝑒−𝜆𝑥 

 (8) 

 (9) 

Now, from substituting equations (8) and (9) in (6) and (7), we have obtained the CDF, PDF of 

[0,1] Truncated Nadarajah-Haghighi Exponential distribution, as follows: 

 

 𝐹(𝑥)[0,1]𝑇𝑁𝐻𝐸 =  
1 −  𝑒1−[1+𝑏(1−𝑒−𝜆𝑥)]

𝑎

1 − 𝑒1−[1+𝑏]𝑎  

 

𝑓(𝑥)[0,1]𝑇𝑁𝐻𝐸 =
𝑎𝑏𝜆[1 + 𝑏(1 − 𝑒−𝜆𝑥)]𝑎−1 𝑒1−[1+𝑏(1−𝑒−𝜆𝑥)]𝑎

 𝑒−𝜆𝑥

1 −  𝑒1−[1+𝑏]𝑎  

 (10) 

 

 

 (11) 

Then, equations (10), (11) respectivelly represent the CDF and PDF of [0,1] Truncated Nadarajah-

Haghighi Exponentail distribution. 

According to equations (10), (11), we can obtain the survival and hazard functions of [0,1] 

TNHEdistribuion, as follows: 

The survival function of [0,1] TNHE distribution:  

 

 𝑆(𝑥, 𝑎, 𝑏, 𝜆)[0,1]𝑇𝑁𝐻𝐸 =
𝑒1−(1+𝑏(1−𝑒−𝜆𝑥))

𝑎

− 𝑒1−(1+𝑏)𝑎

1 − 𝑒1−(1+𝑏)𝑎  

 

(12) 

Hazard rate function of [0, 1] TNHE distribution is [22]  

 

 ℎ(𝑥, 𝑎, 𝑏, 𝜆)[0,1]𝑇𝑁𝐻𝐸 =  
𝑎𝑏𝜆(1+𝑏(1−𝑒−𝜆𝑥))

𝑎−1
 𝑒

1−(1+𝑏(1−𝑒−𝜆𝑥))
𝑎

 𝑒−𝜆𝑥

𝑒
1−(1+𝑏(1−𝑒−𝜆𝑥))

𝑎

−𝑒1−(1+𝑏)𝑎

  

 

(13) 

   

4. Statistical properties of [0,1] Truncated Nadarajah-Haghighi Exponential distribution  

In this section, we introduce some of important Statistical Properties of [0,1] Truncated Nadarajah-

Haghighi Exponential distribution like mixture representation, quantile function, moments, 

incomplete moments, moment generated function, Rényi entropy, Shannon entropy, 𝑐 − entropy, 

and order statistics. 

 4.1 Mixture Representation  

The mixture representation of the PDF is essential in the derivation of the statistical properties of 

[0,1] Truncated Nadarajah-Haghighi Exponential distribution. The mixture representation on 

the [0,1] Truncated Nadarajah-Haghighi Exponential distribution PDF can be written as follows: 

 

  𝑓(𝑥)[0,1]𝑇𝑁𝐻𝐸 =
𝑎𝑏𝜆(1+𝑏(1−𝑒−𝜆𝑥))

𝑎−1
 𝑒

1−(1+𝑏(1−𝑒−𝜆𝑥))
𝑎

 𝑒−𝜆𝑥

1−𝑒1−(1+𝑏)𝑎      
 

 

Using the expansion of exponential, we get that  



 IHJPAS. 37 (2) 2024 

 366 
 

 

 
 𝑒1−(1+𝑏(1−𝑒−𝜆𝑥))

𝑎

  = ∑
1

𝑠!
(1 − (1 + 𝑏(1 − 𝑒−𝜆𝑥))

𝑎
)

𝑠
∞
𝑠=0     

 

Then  

 

 
 𝑓(𝑥)[0,1]𝑇𝑁𝐻𝐸 =

𝑎𝑏𝜆(1+𝑏(1−𝑒−𝜆𝑥))
𝑎−1

  𝑒−𝜆𝑥

1−𝑒1−[1+𝑏]𝑎 ∑
1

𝑠!
(1 − (1 + 𝑏(1 − 𝑒−𝜆𝑥))

𝑎
)

𝑠
∞
𝑠=0       

 

 

Now, by the generalized binomial theorem, we get that  

 

 
(1 − (1 + 𝑏(1 − 𝑒−𝜆𝑥))

𝑎
)

𝑠
  = ∑ (𝑠

𝑛
)∞

𝑛=0 (−1)𝑛 (1 + 𝑏(1 − 𝑒−𝜆𝑥))
𝑎𝑛

     

 

So that  

 

 
 𝑓(𝑥)[0,1]𝑇𝑁𝐻𝐸 =

𝑎𝑏𝜆  𝑒−𝜆𝑥

1−𝑒1−(1+𝑏)𝑎 ∑ ∑
1

𝑠!
 (𝑠

𝑛
)∞

𝑛=0 (−1)𝑛 (1 + 𝑏(1 − 𝑒−𝜆𝑥))
𝑎𝑛+𝑎−1∞

𝑠=0        

 

Again, by the generalized binomial theorem  

 

 
(1 + 𝑏(1 − 𝑒−𝜆𝑥))

𝑎𝑛+𝑎−1
  = ∑ (𝑎𝑛+𝑎−1

𝑑
)∞

𝑑=0  𝑏𝑑 ((1 − 𝑒−𝜆𝑥))
𝑑

     

 

Then 

 

 
 𝑓(𝑥)[0,1]𝑇𝑁𝐻𝐸 =

𝑎𝑏𝜆  𝑒−𝜆𝑥

1−𝑒1−(1+𝑏)𝑎 ∑ ∑ ∑
(−1)𝑛

𝑠!
 (𝑠

𝑛
)(𝑎𝑛+𝑎−1

𝑑
)∞

𝑑=0
∞
𝑛=0 𝑏𝑑  ((1 − 𝑒−𝜆𝑥))

𝑑∞
𝑠=0        

 

 By the same way, we obtain that 

 

 
((1 − 𝑒−𝜆𝑥))

𝑑
  = ∑ (𝑑

𝑘
)∞

𝑘=0 (−1)𝑘𝑒−𝜆𝑘𝑥      

 

 Then  

 

 
 𝑓(𝑥)[0,1]𝑇𝑁𝐻𝐸 =

𝑎𝑏𝜆  

1−𝑒1−(1+𝑏)𝑎 ∑ ∑ ∑ (𝑑
𝑘

)∞
𝑘=0

1

𝑠!
 (𝑠

𝑛
)∞

𝑛=0 (−1)𝑛+𝑘 ∞
𝑠=0 𝑒−𝜆𝑥(𝑘+1)       

Moreover, this equation can be rewritten as follow: 

 

 

 𝑓(𝑥)[0,1]𝑇𝑁𝐻𝐸 = Ω𝑠,𝑛,𝑘,𝑑 𝑒
−𝜆𝑥(𝑘+1)           (14) 

 

Where 

 

 
 Ω𝑠,𝑛,𝑘,𝑑 =  

𝑎𝑏𝜆  

1−𝑒1−(1+𝑏)𝑎 ∑ ∑ ∑ (𝑑
𝑘

)∞
𝑘=0

1

𝑠!
 (𝑠

𝑛
)∞

𝑛=0 (−1)𝑛+𝑘 ∞
𝑠=0        

4.2 Quantile function 

The [0,1] Truncated Nadarajah-Haghighi Exponential distribution quantile function can be 

obtained by inverting the CDF which is defined in (10), as follow [23]: 

 𝑄(𝑢) =  𝐹−1
[0,1]𝑇𝑁𝐻𝐸𝑥𝑝𝑜 (𝑥)  

 

  𝑄(𝑢) = 𝐺−1 (
−1

𝜆
(ln {1 −

1

𝑏
((1 − ln{1 −  𝑢(1 − 𝑒1−(1+𝑏)𝑎

)})
1

𝑎 − 1)}))     
 

(15) 

Now, according to eq. (15), the median (𝑀) of [0.1] TNHE distribution can be obtained by instead 

u = 0.5. 

4.3 Moments  

The 𝑟𝑡ℎ moments of [0,1] Truncated Nadarajah-Haghighi Exponential distribution is [24], [25]. 

 

 
 𝜇𝑟 =  Ω𝑠,𝑛,𝑘,𝑑  (

1

𝜆(𝑘+1)
)

𝑟+1

 Γ(𝑟 + 1)      
 

We can obtain 𝑟𝑡ℎ moment of a random variable X given by the following relation: 

 

 

 𝜇𝑟 =  ∫ 𝑥𝑟𝑓(𝑥)𝑑𝑥
∞

−∞
        



 IHJPAS. 37 (2) 2024 

 367 
 

Where 𝑓(𝑥) is given in (14), so that 

 

 

 𝜇𝑟 =  ∫ 𝑥𝑟Ω𝑠,𝑛,𝑘,𝑑 𝑒
−𝜆𝑥(𝑘+1)𝑑𝑥

∞

0
        

Let 𝑦 =  𝜆𝑥(𝑘 + 1)  ⟹ 𝑥 =  𝑦
1

𝜆(𝑘+1)
  ⟹ 𝑑𝑥 =

1

𝜆(𝑘+1)
 𝑑𝑦   

That is,  

 

 
  𝜇𝑟 =  Ω𝑠,𝑛,𝑘,𝑑 ∫ (𝑦

1

𝜆(𝑘+1)
)

𝑟

  𝑒− 𝑦  
1

𝜆(𝑘+1)
 𝑑𝑦

∞

 0
 

       = Ω𝑠,𝑛,𝑘,𝑑 (
1

𝜆(𝑘+1)
)

𝑟+1

∫ (𝑦)𝑟  𝑒− 𝑦 𝑑𝑦
∞

 0
 

 

 

(16) 

The mean and variance of [0,1] Truncated Nadarajah-Haghighi Exponential distribution can be 

obtained in equation (15), as follow:  

 

 
 𝜇1 = 𝐸(𝑋) [0,1]𝑇𝑁𝐻𝐸 = Ω𝑠,𝑛,𝑘,𝑑 ( 

−1

𝜆(𝑘+1)
)

2

   

 

𝜇2 = 𝐸(𝑋2) [0,1]𝑇𝑁𝐻𝐸 = Ω𝑠,𝑛,𝑘,𝑑
−2

(𝜆(𝑘+1))
3   

 

𝜇3 = 𝐸(𝑋3) [0,1]𝑇𝑁𝐻𝐸 = Ω𝑠,𝑛,𝑘,𝑑
6

(𝜆(𝑘+1))
4  

 

𝜇4 = 𝐸(𝑋4) [0,1]𝑇𝑁𝐻𝐸 = Ω𝑠,𝑛,𝑘,𝑑
−24

(𝜆(𝑘+1))
5  

 

(17)   

                   

          

 

 

(18) 

 

 

(19) 

 

 

(20) 

Moreover, the variance can be found using the following form:  

 

 
 𝑉𝑎𝑟(𝑋) = 𝐸(𝑋2) −  (𝐸(𝑋))

2
        (21) 

So that 

 

 
𝑉𝑎𝑟(𝑋)[0,1]𝑇𝑁𝐻𝐸 = 𝐷𝑗 ,𝑘,𝑚

−2

(𝜆(𝑘+1))
3 − 𝐷𝑗 ,𝑘,𝑚

1

(𝜆(𝑘+1))
4         (22) 

In addition, measures of skewness and kurtosis of [0,1] TNHE distribution based on the above 

equations can be obtained according to the following relations:  

 

 
𝑆𝑘 =  

𝜇3−3𝜇2𝜇1+2𝜇1
3

(𝜇2−𝜇1
2)

3/2          , 𝐾𝑢 =  
𝜇4−4𝜇3𝜇1+6𝜇2𝜇1

2−3𝜇1
4

(𝜇2−𝜇1
2)

2          

4.4 Incomplete Moments 

The incomplete moments of [0,1] Truncated Nadarajah-Haghighi Exponential distribution can 

be obtained in the same way of (4.3), as follows: 

 

 

𝑀𝑟(𝑦) =  ∫ 𝑥𝑟𝑓(𝑥)[0,1]𝑇𝑁𝐻𝐸 𝑑𝑥
𝑦

−∞
         

 

 
               = Ω𝑠,𝑛,𝑘,𝑑  

1

(𝜆(𝑘+1)) 𝑟+1
 𝛾(𝑟 + 2, 𝜆𝑦(𝑚 + 1))                                       (23)   

4.5 Moment Generating Functions  

The Moment Generating Functions of [0,1] Truncated Nadarajah-Haghighi Exponential 

distribution can be obtained in the following: 



 IHJPAS. 37 (2) 2024 

 368 
 

 

 
𝑀𝑋(𝑡) = ∑

𝑙𝑟

𝑙!

∞
𝑙=0 ∫ 𝑥𝑙𝑓(𝑥)[0,1]𝑇𝑁𝐻𝐸 𝑑𝑥

∞

−∞
  

               = ∑
𝑙𝑟

𝑙!

∞
𝑙=0 Ω𝑠,𝑛,𝑘,𝑑 ∫ 𝑥𝑙𝑒−𝜆(𝑘+1)𝑥 𝑑𝑥

∞

0
  

 

 

(24) 

4.6 Entropy  

In this sub-section, we will find three common measures of entropy for a random variable X. 

These are Rényi entropy, Shannon entropy and delta entropy. Shannon entropy is a special case of 

Rényi entropy. 

 

 4.6.1 Rényi entropy 

Rényi entropy of [0,1] Truncated Nadarajah-Haghighi Exponential distribution is defined as 

follows  

 

 
𝐼𝑅(𝑐)[0,1]𝑇𝑁𝐻𝐸 =  

1

1−𝑐
log{∫ 𝑓𝑐(𝑥)𝑑𝑥

∞

−∞
} , 𝑐 ≠ 1 , 𝑐 > 0       

The Rényi entropy for the random variable X is defined by  

 

 
𝐼𝑅(𝑐)[0,1]𝑇𝑁𝐻𝐸 =  

1

1−𝑐
log{∫ 𝑓𝑐(𝑥)𝑑𝑥

∞

−∞
} , 𝑐 ≠ 1 , 𝑐 > 0       

Now 

 

 𝑓𝑐(𝑥) =   (
𝑎𝑏𝜆(1+𝑏(1−𝑒−𝜆𝑥))

𝑎−1
 𝑒

1−(1+𝑏(1−𝑒−𝜆𝑥))
𝑎

 𝑒−𝜆𝑥

1+𝑏
)

𝑐

  

           =  
(𝑎𝑏𝜆)𝑐 𝑒−𝑐𝜆𝑥

(1+𝑏)𝑐  ((1 + 𝑏(1 − 𝑒−𝜆𝑥))
𝑐(𝑎−1)

 𝑒𝑐(1−(1+𝑏(1−𝑒−𝜆𝑥))
𝑎

)) 

 

 Now, by use the expansion exponential formula, we get  

 

 
𝑒𝑐(1−(1+𝑏(1−𝑒−𝜆𝑥))

𝑎
)
 = ∑

1

𝑗!
 𝑐𝑗(1 − (1 + 𝑏(1 − 𝑒−𝜆𝑥))

𝑎
)

𝑗
∞
𝑗=0       

 

So that 

 

 
 𝑓𝑐(𝑥) =

(𝑎𝑏𝜆)𝑐 𝑒−𝑐𝜆𝑥

(1+𝑏)𝑐  ((1 + 𝑏(1 − 𝑒−𝜆𝑥))
𝑐(𝑎−1)

 ∑
1

𝑗!
 𝑐𝑗(1 − (1 + 𝑏(1 − 𝑒−𝜆𝑥))

𝑎
)

𝑗
∞
𝑗=0 ) 

 

And by the generalized binomial theorem  

 

 
 (1 − (1 + 𝑏(1 − 𝑒−𝜆𝑥))

𝑎
)

𝑗
=  ∑ (𝑗

𝑘
)(−1)𝑘(1 + 𝑏(1 − 𝑒−𝜆𝑥))

𝑎𝑗∞
𝑘=0  

 

Hence  

 

 
 𝑓𝑐(𝑥) =

(𝑎𝑏𝜆)𝑐 𝑒−𝑐𝜆𝑥

(1+𝑏)𝑐
 ∑  ∑ (𝑗

𝑘
)

𝑐𝑗

𝑗!
(−1)𝑘(1 + 𝑏(1 − 𝑒−𝜆𝑥))

𝑎𝑗+𝑐𝑎−𝑐∞
𝑘=0

∞
𝑗=0  

 

In the same way  

 

 
 (1 + 𝑏(1 − 𝑒−𝜆𝑥))

𝑎𝑗+𝑐𝑎−𝑐
=  ∑ (𝑎𝑗+𝑐𝑎−𝑐

𝑟
)𝑏𝑟 (1 − 𝑒−𝜆𝑥)𝑟∞

𝑟=0   

Then  

 

 
 𝑓𝑐(𝑥) =

(𝑎𝑏𝜆)𝑐 𝑒−𝑐𝜆𝑥

(1+𝑏)𝑐  ∑  ∑ ∑ (𝑗
𝑘

)
𝑐𝑗

𝑗!
(−1)𝑘(𝑎𝑗+𝑐𝑎−𝑐

𝑟
)𝑏𝑟 (1 − 𝑒−𝜆𝑥)𝑟∞

𝑟=0
∞
𝑘=0

∞
𝑗=0  

 

 

Also  

 

 

 (1 − 𝑒−𝜆𝑥)𝑟 = ∑ (𝑟
𝑠
)(−1)𝑠𝑒−𝑠𝜆𝑥∞

𝑠=0    

 

 



 IHJPAS. 37 (2) 2024 

 369 
 

Finally  

 

 

 𝑓𝑐(𝑥) =  𝛹𝑗,𝑘,𝑟,𝑠 𝑒 −𝜆𝑥(𝑠+𝑐)    

Where 

 

 
 Ψ𝑗,𝑘,𝑟,𝑠 =

(𝑎𝑏𝜆)𝑐 

(1+𝑏)𝑐
 ∑  ∑ ∑  ∑ (𝑟

𝑠
)(𝑗

𝑘
)(𝑎𝑗+𝑐𝑎−𝑐

𝑟
) 

𝑐𝑗

𝑗!
 (−1)𝑘+𝑠 𝑏𝑟 ∞

𝑠=0
∞
𝑟=0

∞
𝑘=0

∞
𝑗=0  

 

Now  

 

 
𝐼𝑅(𝑐)[0,1]𝑇𝑁𝐻−𝐸𝑥𝑝𝑜 =  

1

1−𝑐
log{∫ 𝛹𝑗,𝑘,𝑟,𝑠 𝑒 −𝜆𝑥(𝑠+𝑐) 𝑑𝑥

∞

0
} , 𝑐 ≠ 1 , 𝑐 > 0   

To find 

 

 
𝛹𝑗,𝑘,𝑟,𝑠 ∫  𝑒 −𝜆𝑥(𝑠+𝑐) 𝑑𝑥

∞

0
 = 𝛹𝑗,𝑘,𝑟,𝑠  

1

𝜆(𝑠+𝑐)
  

Then the final form of Rényi entropy for [0,1] TNHE distribution is  

 

 
𝐼𝑅(𝑐)[0,1]𝑇𝑁𝐻−𝐸𝑥𝑝𝑜 =  

1

1−𝑐
log {𝛹𝑗,𝑘,𝑟,𝑠  

1

𝜆(𝑠+𝑐)
} , 𝑐 ≠ 1 , 𝑐 > 0  (25) 

 4.6.2 Shannon entropy 

The Shannon entropy of the new distribution is given by  

 

 

𝜂𝑥 = 𝐸(− log Ω𝑠,𝑛,𝑘,𝑑 𝑒
−𝜆𝑥(𝑘+1))  (26) 

Shannon entropy, defined as an a random variable X with a PDF 𝑓(𝑥), is a special case of the 

Renyi entropy when c↑ 1  and is defined as follow  

    𝜂𝑥 = 𝐸(− log 𝑓(𝑥))  

So, the [0,1]TNHE distribution random variable is given by  

    𝜂𝑥 = 𝐸(− log Ω𝑠,𝑛,𝑘,𝑑 𝑒
−𝜆𝑥(𝑘+1))                                                

4.6.3 Delta Entropy  

The 𝑐 − entropy of a random variable X is given by  

   𝐻(𝑐) =
1

1−𝛿
log{1 − ∫ 𝑓𝑐(𝑥)𝑑𝑥

∞

−∞
}    

Hence, [0,1] Truncated Nadarajah-Haghighi Exponential distribution is given by: 

 𝐻(𝑐) =
1

1−𝑐
log{1 − 𝛹𝑗,𝑘,𝑟,𝑠 ∫   𝑒 −𝜆𝑥(𝑠+𝑐)𝑑𝑥

∞

0
}      

    𝐻(𝛿) =
1

1−𝛿
log {1 − 𝛹𝑗,𝑘,𝑟,𝑠  

1

𝜆(𝑠+𝑐)
}   

4.7 Order statistic 

Let 𝑋1, 𝑋2 , 𝑋3 , … , 𝑋𝑛 have [0,1] Truncated Nadarajah-Haghighi Exponential distribution with 

CDF, PDF defined in (10), (11), respectively and let 𝑋1:𝑛, 𝑋2:𝑛 , 𝑋3:𝑛 , … , 𝑋𝑛:𝑛 be the order statistic 

obtained from this sample. Then, the probability density function of 𝑝𝑡ℎ order statistic from 

[0,1]TNHE distribution is obtained as follows [26]: 

The PDF of order statistic with order 𝑝, 𝑋𝑝;𝑛 is given by the form: 

 

 
𝑓𝑝:𝑛(𝑥) =

𝑛!

(p−1)!(𝑛−𝑝)!
 [𝐹(𝑥)]𝑝−1[1 − 𝐹(𝑥)]𝑛−𝑝 𝑓(𝑥)  

            =  ∑ 𝑑(−1)𝑠(𝑛−𝑝
𝑠

)[𝐹(𝑥)]𝑝+𝑠−1𝑛−𝑝
𝑠=0  𝑓(𝑥) 

 

 

(27) 

Where 𝑑 =
𝑛!

(p−1)!(𝑛−𝑝)!
 

Now, substituting (10) , (11) in (27), we have get 



 IHJPAS. 37 (2) 2024 

 370 
 

 

 𝑓𝑝:𝑛(𝑥) =  ∑ 𝑑 (−1)𝑠 (
𝑛 − 𝑝

𝑠
) (

1 − 𝑒1−(1+𝑏(1−𝑒−𝜆𝑥))   

1 − 𝑒1−(1+𝑏)𝑎 )

𝑝+𝑠−1𝑛−𝑝

𝑠=0

 

              × (
𝑎𝑏𝜆(1+𝑏(1−𝑒−𝜆𝑥))

𝑎−1
 𝑒

1−(1+𝑏(1−𝑒−𝜆𝑥))
𝑎

 𝑒−𝜆𝑥

1−𝑒1−(1+𝑏)𝑎 ) 

 

(28) 

Now, for 𝑝 = 1, we get the smallest order statistic (least value function): 

 

 𝑓𝑝:𝑛(𝑥) =  ∑ 𝑑 (−1)𝑠 (
𝑛 − 1

𝑠
) (

1 − 𝑒1−(1+𝑏(1−𝑒−𝜆𝑥))   

1 − 𝑒1−(1+𝑏)𝑎 )

𝑠𝑛−1

𝑠=0

 

              × (
𝑎𝑏𝜆(1+𝑏(1−𝑒−𝜆𝑥))

𝑎−1
 𝑒

1−(1+𝑏(1−𝑒−𝜆𝑥))
𝑎

 𝑒−𝜆𝑥

1−𝑒1−(1+𝑏)𝑎 ) 

 

(29) 

  And for 𝑝 = 𝑛, we get the largest order statistic (big value function): 

 

 

𝑓𝑛:𝑛(𝑥) =

 (
1−𝑒

1−(1+𝑏(1−𝑒−𝜆𝑥))
   

1−𝑒1−(1+𝑏)𝑎 )

𝑛+𝑠−1

                                                                                            (30) 

                       × (
𝑎𝑏𝜆(1+𝑏(1−𝑒−𝜆𝑥))

𝑎−1
 𝑒

1−(1+𝑏(1−𝑒−𝜆𝑥))
𝑎

 𝑒−𝜆𝑥

1−𝑒1−(1+𝑏)𝑎 ) 

 

5. Maximum Likelihood Method  

 Assume that ,-1., ,𝑥-2.,…, ,𝑥-𝑛. is a random sample of size n from the [0,1] truncated 

Nadarajah-Haghighi Exponential distribution. The corresponding log-likelihood function is then 

given by [27], [28] : 

𝐿(𝜑\𝑋) =
(𝑎𝑏𝜆)𝑛  𝑒−𝜆 ∑ 𝑥𝑖

𝑛
𝑖=1  ∑ ((1 + 𝑏(1 − 𝑒−𝜆𝑥𝑖)))

𝑎−1
𝑛
𝑖=1  𝑒∑ (1−(1+𝑏(1−𝑒−𝜆𝑥𝑖))

𝑎𝑛
𝑖=1 ) 

(1 − 𝑒1−(1+𝑏)𝑎)𝑛
 

Let   𝑙 =  𝑙𝑜𝑔𝐿(𝜑\𝑋) be the natural logarithm probability function. 

𝑙 = 𝑛𝑙𝑜𝑔(𝑎𝑏𝜆) − 𝜆 ∑ 𝑥𝑖

𝑛

𝑖=1

+ (𝑎 − 1) ∑ 𝑙𝑜𝑔{(1 + 𝑏(1 − 𝑒−𝜆𝑥𝑖))} 

𝑛

𝑖=1

+  ∑(1 − (1 + 𝑏(1 − 𝑒−𝜆𝑥𝑖))
𝑎

𝑛

𝑖=1

) − 𝑛𝑙𝑜𝑔{1 − 𝑒1−(1+𝑏)𝑎
} 

𝑙 = 𝑛𝑙𝑜𝑔(𝑎) + 𝑛𝑙𝑜𝑔(𝑏) + 𝑛𝑙𝑜𝑔(𝜆) − 𝜆 ∑ 𝑥𝑖

𝑛

𝑖=1

+ (𝑎 ∑ 𝑙𝑜𝑔 {(1 + 𝑏(1 − 𝑒−𝜆𝑥𝑖))} 

𝑛

𝑖=1

− ∑ 𝑙𝑜𝑔 {(1 + 𝑏(1 − 𝑒−𝜆𝑥𝑖))} 

𝑛

𝑖=1

) + ∑ 1

𝑛

𝑖=1

− ∑(1 + 𝑏(1 − 𝑒−𝜆𝑥𝑖))
𝑎

𝑛

𝑖=1

) − 𝑛𝑙𝑜𝑔(1 − 𝑒1−(1+𝑏)𝑎
) 



 IHJPAS. 37 (2) 2024 

 371 
 

𝑙 = 𝑛𝑙𝑜𝑔(𝑎) + 𝑛𝑙𝑜𝑔(𝑏) + 𝑛𝑙𝑜𝑔(𝜆) − 𝜆 ∑ 𝑥𝑖

𝑛

𝑖=1

+ (𝑎 ∑ 𝑙𝑜𝑔{(1 + 𝑏(1 − 𝑒−𝜆𝑥𝑖))} 

𝑛

𝑖=1

− ∑ 𝑙𝑜𝑔{(1 + 𝑏(1 − 𝑒−𝜆𝑥𝑖))} 

𝑛

𝑖=1

) +  𝑛

− ∑(1 + 𝑏(1 − 𝑒−𝜆𝑥𝑖))
𝑎

𝑛

𝑖=1

) − 𝑛𝑙𝑜𝑔{1 − 𝑒1−(1+𝑏)𝑎
} 

Now, by taking the first partial derivative of the log likelihood function with respect to the 

parameters (𝑎, 𝑏, 𝜆), we get: 

𝜕(𝑙)

𝜕𝑎
= {

𝑛

𝑎
+  ∑ 𝑙𝑜𝑔{(1 + 𝑏(1 − 𝑒−𝜆𝑥𝑖))}

𝑛

𝑖=1

+  (1 + 𝑏(1 − 𝑒−𝜆𝑥𝑖))
𝑎

 𝑙𝑜𝑔{(1 + 𝑏(1 − 𝑒−𝜆𝑥𝑖))
𝑎

}   

−   
 𝑛 (1 + 𝑏)𝑎  𝑙𝑜𝑔 {1 + 𝑏}𝑒1−(1+𝑏)𝑎 

1 − 𝑒1−(1+𝑏)𝑎      } 

 

 

 

 

(31) 

 

𝜕(𝑙)

𝜕𝑏
=

𝑛

𝑏
+ (𝑎 − 1) ∑

1 − 𝑒−𝜆𝑥𝑖

1 + 𝑏(1 − 𝑒−𝜆𝑥𝑖)

𝑛

𝑖=1

− 

        ∑ (𝑎(1 − 𝑒−𝜆𝑥𝑖) (1 + 𝑏(1 − 𝑒−𝜆𝑥𝑖))
𝑎−1

)

𝑛

𝑖=1

−
𝑛𝑎(1 + 𝑏)𝑎−1   𝑒1−(1+𝑏)𝑎 

1 − 𝑒1−(1+𝑏)𝑎  

 

 

 

 

(32) 

𝜕(𝑙)

𝜕𝜆
=

𝑛

𝜆
−  ∑ 𝑥𝑖

𝑛
𝑖=1 + (𝑎 − 1) ∑

𝑏 𝑥𝑖 𝑒−𝜆𝑥𝑖

1+𝑏(1−𝑒−𝜆𝑥𝑖)
−  ∑  𝑎𝑏 𝑥𝑖  𝑒−𝜆𝑥𝑖  (1 + 𝑏(1 −𝑛

𝑖=1
𝑛
𝑖=1

𝑒−𝜆𝑥𝑖))
𝑎−1

   

(33) 

By setting the above equations (31), (32), and (33) to zero, solving it numerically through  using 

iterative methods, such as Newton-Raphson type algorithms, we can get the estimators of the 

parameters. 

 

6. Simulation Study 

  In this section, we have conducted simulation study for [0,1] TNHE distribution. We have 

generated samples of sizes 𝑛 = {30, 50, 80, 120, 200}  from the proposed model and parameters 

estimated by MLE method, the simulation study is in terms of the averages of the three quantities: 

absolute bias |Bais(𝝉)| =
1

N
∑ |N

i=1 �̂� − 𝝉|, mean square error (MSE), 𝑀𝑆𝐸(𝝉) =
1

N
∑ (�̂� − 𝝉)2N

i=1 , 

and the mean relative error (MRE), 𝑀𝑅𝐸(𝝉) =
1

N
∑ |N

i=1 �̂� − 𝝉|/𝝉 . All the computations are made 

by using R Statistical Software. Table 1 shows some simulation results for different values of 

𝝉 = (𝑎, 𝑏, 𝜆)𝑇. Based on the simulation criteria, Table 1 discovered that the maximum likelihood 

estimate strategy performs pretty well in estimating the [0,1] TNHE distribution parameters. 

 

7.  Application 

  In this section, we fit the TNHE distribution to a real data set to show that the proposed 

distribution fits well compared to competing distributions. The statistical software R is used to 

calculate all  results. To obtain the best results, we used the following statistical criteria (-𝑙, AIC, 

AIC, BIC, HQIC) for the proposed model compared to other models such as Beta-Exponential 



 IHJPAS. 37 (2) 2024 

 372 
 

(BeEx), Kumaraswamy-Exponential (KuEx), Exponential Generalized Exponential (EGEx), 

Weibull-Exponential (WeEx), Gompertez-Exponential (GoEx), Marshal-Olkin-Ex (MoEx) and 

Exponential (Ex). This data set is the use of the failure rate data set (103 hours) for the turbocharger 

for the engine type. For the dataset, we consider the large recorded intensities (on the Richter scale) 

of earthquakes at seismometer locations in western North America between 1940 and 1980, as in 

[29], [30], [31].  

7.5,8.8,8.9,9.4,9.7,9.7,10.5,10.5,12,12.2,12.8,14.6,14.9,17.6,23.9,25,2.9,3.2,7.6,17,8,10,10,8,1

9,21,13,22,29,31,5.8,12,12.1,20.5,20.5,25.3,35.9,36.1,36.3,38.5,41.4,43.6,44.4,46.1,47.1,47.7,

49.2,53.1,4,10.1,11.1,17.7,22.5,26.5,29,30.9,37.8,48.3,62,50,16,62,1.2,1.6,9.1,3.7,5.3,7.4,17.9,

19.2,23.4,30,38.9,10.8,15.7,16.7,20.8,28.5,33.1,40.3,8,32,30,31,16.1,63.6,6.6,9.3,13,17.3,105,

112,123,5,23.5,26,0.5,0.6,1.3,1.4,2.6,3.8,4,5.1,6.2,6.8,7.5,7.6,8.4,8.5,8.5,10.6,12.6,12.7,12.9,1

4,15,16,17.7,18,22,22,23,23.2,29,32,32.7,36,43.5,49,60,64,105,122,141,200,45,130,147,187,1

97,203,211,17,19.6,20.2,21.1,21.9,66,87,23.4,24.6,25.7,28.6,37.4,46.7,56.9,60.7,61.4,62,64,82

,88,91,12,24.2,148,42,85,107,109,156,224,293,359,370,25.4,32.9,92.2,45,145,300. 

According to the values shown in Tables 2 and 3, it is clear that the TNHE distribution is 

superior to the comparative distributions. The proposed expanded distribution provides an 

accurate representation because it has the lowest values according to the statistical and 

informational criteria and the largest value of the𝑝-value. It is from Figures 1 and 2, the [0,1] 

TNHE model provides the overall best fit and therefore could be chosen as the adequate model 

for explaining data. 

 

Table 1. Bias, MSE and MRE of parameters of [0,1] TNHE distribution. 

τ =  ( a =  0.75, b =  1.75, λ =  3)T 

𝑛 = 200 𝑛 = 120 𝑛 = 80 𝑛 = 50 𝑛 = 30 Est. Par. Est. 

0.61788  0.68144  0.73878  0.80613 0.87476  �̂� |Bias| 

0.92474  1.17777  1.38657  1.66894  1.96507  �̂�  

0.55274  0.67574  0.78722  0.90548  1.08240  𝜆 ̂  

0.56899  0.71129  0.87907  1.02999  1.23050  �̂� MSE 

1.48122  2.61345  3.67168  5.59836  7.99706  �̂�  

0.47828  0.72033  0.98659  1.34383  1.94217  𝜆 ̂  

0.82383  0.90858  0.98504  1.07483  1.16634  �̂� MRE 

0.52842  0.67301  0.79233  0.95368  1.12290  �̂�  

0.18425  0.22525  0.26241  0.30183  0.36080  𝜆 ̂  

𝜏 = (a =  1.5, b =  0.5, λ  =  0.4)T 

𝑛 = 200 𝑛 = 120 𝑛 = 80 𝑛 = 50 𝑛 = 30 Est. Par. Est. 

0.60516  0.62884  0.64801  0.66331  0.69942  �̂� |Bias| 

0.50648  0.61350  0.71024  0.81820  0.91634  �̂�  

0.06481  0.07838  0.08630  0.09739  0.11670  𝜆 ̂  

0.74607  0.79344  0.82684  0.8509  0.93774  �̂� MSE 

0.44504  0.63727  0.81895  1.07283  1.34474  �̂�  

0.00685  0.00962  0.01156  0.01483  0.02236  𝜆 ̂  

0.40344  0.41923  0.43201  0.44220  0.46628  �̂� MRE 

1.01296  1.22700  1.42047  1.63640  1.83267  �̂�  

0.16203  0.19595  0.21574  0.24347  0.29175  𝜆 ̂  

𝜏 = (a =  3, b =  3, λ =  1.6)𝑇 

𝑛 = 200 𝑛 = 120 𝑛 = 80 𝑛 = 50 𝑛 = 30 Est. Par. Est. 

0.74159  0.94522 1.11533  1.31950 1.62784  �̂� |Bias| 

0.43437  0.48626  0.54433  0.60525  0.67291  �̂�  

0.38669  0.46325  0.53197  0.62328  0.74430  𝜆 ̂  

0.93697  1.51047  2.05816  2.85928  4.13035  �̂� MSE 

0.31474  0.40259  0.50889  0.64676  0.80849  �̂�  

0.22417  0.32020  0.42608  0.63751  1.0537  𝜆 ̂  



 IHJPAS. 37 (2) 2024 

 373 
 

 

Table 2.  The K-S value with its corresponding 𝑝-value and W value of the data set 

value 

with 

icorrue of the  
 

Table 3.  Represented the values of statistically criteria (-LL, AIC, CAIC, BIC, HQIC). 

 

 

 

 

0.24720  0.31507  0.37178  0.43983  0.54261  �̂� MRE 

0.14479  0.16209  0.18144  0.20175  0.22430  �̂�  

0.24168  0.28953  0.33248  0.38955  0.46519  𝜆 ̂  

Model W A K-S 𝒑-value 

[0,1]TNH-Expo        0.2156     1.2964  0.08246 0.1682 

BeEx 0.5323 3.0980  0.1149  0.0162 

KuEx 0.5134 2.9906 0.1068 0.0313 

EGEx  0.53635 3.1207 0.1169 0.0138 

WeEx 0.2172 1.3341 0.0908 0.0994 

GoEx 0.2172 1.3341 0.0773 0.2259 

MoEx  0.5079  2.9607 0.1218 0.0090 

Ex  0.5289 3.0797  0.1267 0.0057 

Model MLEs - 𝒍 AIC CAIC BIC HQIC 

[0,1] TNHE 

�̂�  = 1.154 

�̂�  = 2.388 

�̂�  = 0.010 

 

868.143 

 

 

1742.286  

 

1742.421  

 

1751.898  

 

1746.183 

BeE 

�̂�  = 0.903 

�̂� = 1.482 

�̂� = 0.013 

 

876.633 

 

 

1759.266  

 

1759.401  

 

1768.878  

 

1763.162 

KuE 

�̂� = 0.892 

�̂� = 2.327 

�̂� = 0.008 

 

876.051 

 

 

1758.103  

 

1758.238  

 

1767.715  

 

1762 

EGE 

�̂� = 1.667 

�̂� = 0.911 

�̂� = 0.012 

 

876.782 

 

 

1759.565  

 

1759.7  

 

1769.177  

 

1763.462 

WeE 

�̂� = 0.883 

�̂� = 0.259 

�̂� = 0.006 

 

874.402 

 

 

1745.208  

 

1745.343  

 

1754.82  

 

1749.105 

GoE 

�̂�  = 1.006 

�̂�  = 0.171 

�̂�  = 0.028 

869.604 

 

 

1745.208  

 

1745.343  

 

1754.82  

 

1749.105 

MoE 
�̂�  = 0.944 

�̂�  = 0.021 

 

876.340 

 

 

1756.683  

 

1756.75  

 

1763.091  

 

1759.281 

E �̂�  = 0.021 

 

877.236 

 

  

1756.473  

 

 1756.495  

  

1759.677  

 

 1757.772 



 IHJPAS. 37 (2) 2024 

 374 
 

 
    

Figure 1. Estimated fitted densities of model for dataset. 

 

 
Figure 2. Estimated fitted CDF for data set. 



 IHJPAS. 37 (2) 2024 

 375 
 

 
 

Figure 3. TTT plot of [0,1] TNHE distribution for data set. 

 

5. Conclusion 

This paper proposes a new extension of the exponential distribution based on the [0,1] Truncated 

Nadarajah-Haghighi-G family of distributions called [0,1] Truncated Nadarajah-Haghighi. The 

exponential distribution, which is a new distribution with three parameters, is more flexible than 

some other distributions, such as the exponential distribution and the Weibull exponential 

distribution. Also, we derive some statistical properties for the new distribution, such as the 

quantile function, moments, incomplete moments and entropy. Finally, we use the maximum 

likelihood method to estimate the parameters for the new distribution. 

 

Acknowledgment 

Our researcher extends his Sincere thanks to the editor and members of the preparatory committee 

of the Ibn AL-Haitham Journal of Pure and Applied Sciences. 

 

Conflict of Interest  

There are no conflicts of interest. 

 

Funding  

There is no funding for the article. 

 

References 

1. Eugene, N; Lee, C; Famoye, F. Beta-normal distribution and its applications Communications in 

Statistics-Theory and methods 2002, 31(4), 497-512 https://doi.org/10.1081/STA-120003130 

2. Cordeiro, G. M; Ortega, E. M; da Cunha, D. C. The exponentiated generalized class of distributions. 

Journal of data science 2013, 11(1), 1-27. https://doi.org/10.6339/JDS.2013.11%281%29.1086 

3. Bourguignon M; Silva RB; Cordeiro GM.  The Weibull-G family of probability distribution. Journal of 

Data Science 2014, 12(1), 53-68. https://doi.org/10.6339/JDS.2014.12(1).1210 

https://doi.org/10.6339/JDS.2013.11%281%29.1086
https://doi.org/10.6339/JDS.2014.12(1).1210


 IHJPAS. 37 (2) 2024 

 376 
 

4. Nofal, Z. M; Afify, A. Z; Yousof, H. M; Cordeiro, G. M. The generalized transmuted-G family of 

distributions. Communications in Statistics-Theory and Methods 2017, 46(8), 4119-4136. 

https://doi.org/10.1080/03610926.2015.1078478 

5. Alizadeh, M; Cordeiro, G. M; Pinho, L. G. B; Ghosh, I. The Gompertz-G family of distributions. Journal 

of statistical theory and practice 2017, 11, 179-207. https://doi.org/10.1080/15598608.2016.1267668 

6. Afify, A. Z; Cordeiro, G. M; Maed, M. E; Alizadeh, M; Al-Mofleh, H; Nofal, Z. M. The generalized 

odd Lindley-G family: Properties and applications. Anais da Academia Brasileira de Ciências 2019, 91. 

https://doi.org/10.1590/0001-3765201920180040 

7. Hosseini, B; Afshari, M; Alizadeh, M. The generalized odd gamma-g family of distributions: properties 

and applications. Austrian Journal of Statistics 2018, 47(2), 69-89. 

https://doi.org/10.17713/ajs.v47i2.580 

8. Nassar, M; Kumar, D; Dey, S; Cordeiro, G. M; Afify, A. Z. The Marshall–Olkin alpha power family of 

distributions with applications. Journal of Computational and Applied Mathematics 2019, 351, 41-53. 

https://doi.org/10.1016/j.cam.2018.10.052 

9. Arshad, R. M. I; Tahir, M. H; Chesneau, C; Jamal, F. The Gamma Kumaraswamy-G family of 

distributions: theory, inference and applications. Statistics in Transition new series 2020, 21(5), 17-40. 

https://doi.org/10.21307/stattrans-2020-053 

10. Khaleel, M. A; Oguntunde, P. E; Al Abbasi, J. N; Ibrahim, N. A; AbuJarad, M. H. The Marshall-Olkin 

Topp Leone-G family of distributions: A family for generalizing. Probability moels. Scientific African 

2020, 8, e00470. https://doi.org/10.1016/j.sciaf.2020.e00470 

11. Afify, A. Z; Al-Mofleh, H; Aljohani, H. M; Cordeiro, G. M. The Marshall–Olkin Weibull-H family: 

estimation, simulations, and applications to COVID-19 data. Journal of King Saud University-Science 

2022, 34(5), 102115. https://doi.org/10.1016/j.jksus.2022.102115 

12. Anzagra, L; Sarpong, S; Nasiru, S. Odd Chen-G family of distributions. Annals of Data Science 2022,  

9(2), 369-391. https://link.springer.com/article/10.1007/s40745-020-00248-2 

13. Abid, S. H; Kadhim, F. J. [0, 1] truncated inverted Gamma-Inverted gamma distribution. In AIP 

Conference Proceedings 2022, 2394(1), 070038. http://dx.doi.org/10.1063/5.0121500 

14. Khaleel, M. A; Abdulwahab, A. M; Gaftan, A. M; Abdal-hammed, M. K. A new [0, 1] truncated inverse 

Weibull rayleigh distribution properties with application to COVID-19. International Journal of 

Nonlinear Analysis and Applications 2022, 13(1), 2933-2946. 

https://doi.org/10.22075/ijnaa.2022.6026 

15. Khalaf, A. A; Khaleel, M. A. [0, 1] Truncated exponentiated exponential gompertz distribution: 

Properties and applications. In AIP Conference Proceedings 2022, 2394(1), 070035. AIP Publishing 

LLC. http://dx.doi.org/10.1063/5.0121072 

16. Gupta, R. D; Kundu, D. Generalized exponential distribution: different method of estimations. Journal 

of Statistical Computation and Simulation 2001, 69, 315-337. 

https://doi.org/10.1080/00949650108812098 

17. Nassar, M; Dey, S; Kumar, D. A new generalization of the exponentiated Pareto distribution with an 

application. Amer. J. Math. Manag. Sci 2018c, 37, 217-242. 

http://dx.doi.org/10.1080/01966324.2017.1396942 

18. Bilal, M; Mohsin, M; Aslam, M. Weibull-exponential distribution and its application in monitoring 

industrial process. Mathematical Problems in Engineering 2021, 1-13. 

https://doi.org/10.1155/2021/6650237 

19. Nadarajah, S; ; Kotz, S. The beta exponential distribution. Reliability engineering & system safety 

2006, 91(6), 689-697. https://doi.org/10.1016/j.ress.2005.05.008 

20. Rasekhi, M; Alizadeh, M; Altun, E; Hamedani, G. G; Afify, A. Z; Ahmad, M. THE MODIFIED 

EXPONENTIAL DISTRIBUTION WITH APPLICATIONS. Pakistan Journal of Statistics 

2017, 33(5). 

https://doi.org/10.1080/03610926.2015.1078478
https://doi.org/10.1080/15598608.2016.1267668
https://doi.org/10.1590/0001-3765201920180040
https://doi.org/10.17713/ajs.v47i2.580
https://doi.org/10.1016/j.cam.2018.10.052
https://doi.org/10.21307/stattrans-2020-053
https://doi.org/10.1016/j.sciaf.2020.e00470
https://doi.org/10.1016/j.jksus.2022.102115
https://link.springer.com/article/10.1007/s40745-020-00248-2
http://dx.doi.org/10.1063/5.0121500
https://doi.org/10.22075/ijnaa.2022.6026
http://dx.doi.org/10.1063/5.0121072
https://doi.org/10.1080/00949650108812098
http://dx.doi.org/10.1080/01966324.2017.1396942
https://doi.org/10.1155/2021/6650237
https://doi.org/10.1016/j.ress.2005.05.008


 IHJPAS. 37 (2) 2024 

 377 
 

21. Nadarajah, S; Haghighi, F. An extension of the exponential distribution. Statistics 2011. 45(6), 543-

558. https://doi.org/10.1080/02331881003678678 

22. Oguntunde, P. E.; Khaleel, M. A.; Ahmed, M. T.,; Adejumo, A. O., ;  Odetunmibi, O. A. A new 

generalization of the Lomax distribution with increasing, decreasing, and constant failure 

rate. Modelling and Simulation in Engineering, 2017. https://doi.org/10.1155/2017/6043169 

23. ul Haq, M. A.; Handique, L., ; Chakraborty, S. The odd moment exponential family of distributions: 

Its properties and applications. International Journal of Applied Mathematics and Statistics, 2018, 57, 

47–62. 

24. Goerg, G. M. Lambert W random variables—A new family of generalized skewed distributions with 

applications to risk estimation. Annals of Applied Statistics, 2011, 5, 2197–2230. 

https://doi.org/10.48550/arXiv.0912.4554 

25. De Gusmao, F. R.; Ortega, E. M., ; Cordeiro, G. M.  The generalized inverse Weibull distribution, 

Statistical Papers, 2011,  52, 591-619. http://dx.doi.org/10.1007/s00362-009-0271-3 

26. Venegas, O.; Iriarte, Y. A.; Astorga, J. M., ; Gómez, H. W. Lomax-Rayleigh distribution with an 

application. Appl. Math. Inf. Sci, 2019, 13(5), 741-748. http://dx.doi.org/10.18576/amis/130506 

27. Lai, C. D.;  Xie, M., ;  Murthy, D. N. P.. A modified Weibull distribution. IEEE Transactions on 

reliability, 2003, 52(1), 33-37. https://doi.org/10.1109/TR.2002.805788 

28. Khubaz, A. F.; Abdal-Hameed, M. K.; Mohamood, N. H.;  Khaleel, M. A. Gompertz Inverse Weibull 

Distribution, some statistical properties with Application Real Dataset. Tikrit Journal of Administration 

and Economics Sciences, 2023, 19(Special Issue part 5). 

http://www.doi.org/10.25130/tjaes.19.SP3.5.11 

29. Teimouri, M.. MPS: an R package for modelling new families of distributions. arXiv preprint arXiv: 

2018, 1809.02959. https://doi.org/10.48550/arXiv.1809.02959 

30. Hadi, H. H.; Al-Noor, N. H.Truncated exponential Marshall Olkin Lomax distribution: Properties, 

entropies, and applications. In AIP Conference Proceedings 2023, 2414(1).AIP Publishing. 

https://doi.org/10.1063/5.0115489. 

 

https://doi.org/10.1080/02331881003678678
https://doi.org/10.1155/2017/6043169
https://doi.org/10.48550/arXiv.0912.4554
http://dx.doi.org/10.1007/s00362-009-0271-3
http://dx.doi.org/10.18576/amis/130506
https://doi.org/10.1109/TR.2002.805788
http://www.doi.org/10.25130/tjaes.19.SP3.5.11
https://doi.org/10.48550/arXiv.1809.02959
https://doi.org/10.1063/5.0115489