Int. J. Anal. Appl. (2022), 20:23 

 

 

Received: Feb. 14, 2022. 

2010 Mathematics Subject Classification. 65C20. 

Key words and phrases. bounded model; mathematical statistics; probability model; entropy; simulation; unit interval 

data. 

 

https://doi.org/10.28924/2291-8639-20-2022-23 © 2022 the author(s) 

ISSN: 2291-8639  

1 

 

On Some Properties of a New Truncated Model With Applications to Lifetime Data 

 

Muhammad Zeshan Arshad1, Oluwafemi Samson Balogun2,*, Muhammad Zafar Iqbal1, 

Pelumi E. Oguntunde3 

 

1Department of Mathematics and Statistics, University of Agriculture, Faisalabad, Pakistan 

2School of Computing, University of Eastern Finland, Kuopio, Northern Europe, 70211, Finland 

3Department of Mathematics, Covenant University, Ota, Ogun State, Nigeria 

*Corresponding author: samson.balogun@uef.fi 

ABSTRACT. This research explored the exponentiated left truncated power distribution which is a bounded 

model. Various statistical properties which include the moments and their associated measures, Bonferroni 

and Lorenz curves, reliability measures, shapes, quantile function, entropy, and order statistics were discussed 

in detail. A simulation study was provided and applications to two real-world data were considered. The 

performance of the exponentiated left truncated power distribution over other bounded models like Topp-

Leone distribution, Beta distribution, Kumaraswamy distribution, Lehmann type–I distribution, Lehmann 

type–II distribution, generalized power function, Weibull power function, and Mustapha type–II distribution 

is quite commendable. 

 

1. Introduction 

Probability models play important roles in describing real-life events. They have been 

discussed in the past to model several real-time events so proficiently. The rainfall event was 

addressed by [1]. Pollution events were addressed by ([2], [3], [4]). Manifold dynamics of COVID-19 

were addressed by ([5], [6]). Engineering issues were addressed by ([7], [8]), and several others. Some 

https://doi.org/10.28924/2291-8639-20-2022-23


2 Int. J. Anal. Appl. (2022), 20:23 

 

probability models are bounded while some are unbounded. Unbounded distributions extend from 

negative infinity to positive infinity while bounded ones are confined to lie between two determined 

values. According to [9], probability models with unit intervals are useful in the area of biology, 

economics, engineering, and psychology among others. Examples of bounded models include the 

continuous uniform distribution, beta distribution, Kumaraswamy distribution by [10], [11], [12], [13], 

and several other notable ones.  

It is also worthy of note that some of these bounded probability models have been used to 

develop generalized families of distributions, examples include the Beta-G family of distributions by 

[14], Kumaraswamy-G family of distributions by [15], Topp-Leone G family of distributions by ([16], 

[17]), and so on. A quest to develop models that can adequately fit real-life events has led to the 

extension of the existing probability models. 

1.1. Definition  

A random variable X is said to follow the ELTr-PF distribution if the associated cumulative 

distribution function (CDF) and corresponding probability density function (PDF) begin at k, and 

are given respectively by; 

𝐹𝐸𝐿𝑇𝑟−𝑃𝐹(𝑥;𝑎,𝑏) = (
𝑥𝑎 −𝑘𝑎

1−𝑘𝑎
)

𝑏

,𝑘 < 𝑥 < 1,𝑎,𝑏 > 0, (1) 

𝑓𝐸𝐿𝑇𝑟−𝑃𝐹(𝑥;𝑎,𝑏) =
𝑎𝑏

(1−𝑘𝑎)𝑏
 𝑥
𝑎−1(𝑥𝑎 −𝑘𝑎)𝑏−1 , (2) 

where 𝑘 < 𝑥 is a possible minimum assured life, and it can be defined as an unknown starting point 

at which age of some certain component/device initiates, and (𝑎,𝑏 > 0) are two shape parameters. 

However, if parameters b=1 and k=0, the model reduces to the baseline model (𝑥𝑎). This research 

is aimed at extending the power function and introducing a new bounded probability model; the 

exponentiated left truncated power (ELTr-PF) function which can be used as an alternative to the 

existing ones because of its superior modeling capabilities. Its properties are identified, a simulation 

and real-life applications are provided. 

The rest of the paper is structured as follows; general mathematical properties of the ELTr-PF 

distribution including reliability measures are derived in Section 2, its miscellaneous measures are 

established in Section 3. The model parameters are estimated in Section 4 while a simulation 

experiment is performed in Section 5. Applications to real-world data sets are discussed in Section 

6, and finally, the conclusion is reported in Section 7. 

 



3 Int. J. Anal. Appl. (2022), 20:23 

 

2. Mathematical Properties 

This section covers several mathematical properties of the exponentiated left truncated power 

distribution. 

 

2.1. Useful representation  

Linear combination provides a much informal approach to discuss the CDF and PDF than the 

conventional integral computation when determining the mathematical properties. For this, the 

following binomial expansion is considered: 

(1−𝑦)𝛽 = ∑(
𝛽
𝑖
)(−1)𝑖𝑦𝑖

∞

𝑖=0

, |𝑦| < 1. 

Owing to Equations (1) and (2), infinite linear combinations (LC) of the ELTr-PF CDF becomes: 

𝐹𝐿𝐶−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑥;𝑎,𝑏) =
1

(1−𝑘𝑎)𝑏
 ∑ (

𝑏
𝑖
)(−1)𝑖𝑘𝑎𝑖𝑥𝑎(𝑏−𝑖),

∞

𝑖=0
 

    

(3) 

and the corresponding PDF is given as follows: 

𝑓𝐿𝐶−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑥;𝑎,𝑏) =
𝑎𝑏

(1−𝑘𝑎)𝑏
 ∑ (

𝑏 −1
𝑗
)(−1)𝑗𝑘𝑎𝑗𝑥𝑎(𝑏−𝑗)−1.

∞

𝑗=0
 (4) 

 

2.2. Moments with associated measures 

Moments play remarkable roles in the discussion of distribution theory in studying the significant 

characteristics of a probability distribution like the mean, variance, skewness, and kurtosis. 

Theorem 1. If X~ 𝐸𝐿𝑇𝑟 −𝑃𝐹(𝑥;𝑘,𝑎,𝑏), with 𝑎,𝑏 > 0, and k < x, then the r-th ordinary moment 

(μ 𝑟
/
) of X is given by: 

𝜇
 𝑟−𝐸𝐿𝑇𝑟−𝑃𝐹
/

=
𝑎𝑏

(1−𝑘
𝑎
)
𝑏 
∑ (

𝑏−1
𝑗
)
(−1)𝑗𝑘

𝑎𝑗

𝑟+𝑎(𝑏−𝑗)
(1−𝑘

𝑟+𝑎(𝑏−𝑗)
) .

∞

𝑗=0
 

Proof  𝜇 𝑟
/
 can be written directly following Equation (4) as follows: 

𝜇
 𝑟−𝐸𝐿𝑇𝑟−𝑃𝐹
/

=
𝑎𝑏

(1−𝑘
𝑎
)
𝑏 
∫ 𝑥𝑟
1

𝑘
𝑥𝑎−1(𝑥𝑎 −𝑘

𝑎
)
𝑏−1 
𝑑𝑥, 

𝜇
 𝑟−𝐸𝐿𝑇𝑟−𝑃𝐹
/

=
𝑎𝑏

(1−𝑘
𝑎
)
𝑏 
∑ (

𝑏−1
𝑗
)(−1)𝑗𝑘

𝑎𝑗
∫ 𝑥𝑟+𝑎(𝑏−𝑗)−1
1

𝑘
𝑑𝑥 

∞

𝑗=0
. 

 



4 Int. J. Anal. Appl. (2022), 20:23 

 

Further, by solving the simple integral computation, it leads to the final form of the r-th ordinary 

moment, and it is given by: 

𝜇
 𝑟−𝐸𝐿𝑇𝑟−𝑃𝐹
/

=
𝑎𝑏

(1−𝑘
𝑎
)
𝑏 
∑ (

𝑏−1
𝑗
)
(−1)𝑗𝑘

𝑎𝑗

𝑟+𝑎(𝑏−𝑗)
(1−𝑘

𝑟+𝑎(𝑏−𝑗)
) 

∞

𝑗=0
. 

 

(5) 

The expression in Equation (5) is quite impressive and useful in the development of several statistical 

measures. For instance, to obtain the mean of X, substitute r = 1 in Equation (5) as follows: 

𝜇
 1−𝐸𝐿𝑇𝑟−𝑃𝐹
/

=
𝑎𝑏

(1−𝑘
𝑎
)
𝑏 
∑ (

𝑏−1
𝑗
)
(−1)𝑗𝑘

𝑎𝑗

1+𝑎(𝑏−𝑗)
(1−𝑘

1+𝑎(𝑏−𝑗)
) 

∞

𝑗=0
. (6) 

One may perhaps further determine the well-established statistics such as skewness (𝛽1 = 𝜇3
2 𝜇2

3⁄ ), 

and kurtosis (𝛽2 = 𝜇4 𝜇2
2⁄ ), of X by integrating Equation (6). A well-established relationship between 

the central moments (𝜇𝑠) and cumulants (𝐾𝑠) of X may easily be defined by ordinary moments 𝜇𝑠 =

∑ (
𝑠
𝑘
)(−1)𝑘 (𝜇1

/
)
𝑠
𝜇𝑠−𝑘
/𝑠

𝑘=0 . Hence, the first four cumulants can be calculated by 𝐾1 = 𝜇1
/
, 𝐾2 = 𝜇2

/
−

𝜇1
/2

 , 𝐾3 = 𝜇3
/
−3𝜇2

/
𝜇1
/
+2𝜇1

/3
, and 𝐾4 = 𝜇4

/
−4𝜇3

/
𝜇1
/
−3𝜇2

/2
+12𝜇2

/
𝜇1
/2
−6𝜇1

/4
, etc., respectively. 

 

Table 1 presents some numerical results of the first four ordinary moments (𝜇/
1
,𝜇/

2
,𝜇/

3
,𝜇/

4
), 𝜎2 = 

variance, 𝛽1 = skewness, and 𝛽2 = kurtosis for some choices of model parameters (k = 0.1) for the 

ELTr-PF distribution. 

 

Table 1. Some numerical results of moments, variance, skewness, and kurtosis. 

Statistics  𝑎 = 0.1,𝑘 = 0.1 
Remarks 

𝜇/
𝑟 𝑏 = 1.2 𝑏 = 1.3 𝑏 = 1.4 𝑏 = 1.5 𝑏 = 1.6 

𝜇/
1
 0.4425 0.4586 0.4737 0.4880 0.5015 

D
e
c
re

a
si
n
g
 𝜇/

2
 0.2607 0.2752 0.2890 0.3023 0.3151 

𝜇/
3
 0.1814 0.1931 0.2045 0.2155 0.2262 

𝜇/
4
 0.1386 0.1482 0.1576 0.1668 0.1758 

𝜎2 0.0373 0.0314 0.0244 0.0165 0.0078 

𝛽1 0.0016 0.0029 0.0031 0.0021 0.0008 Decreasing 

𝛽2 0.1329 0.1266 0.1150 0.0971 0.0714 Decreasing 

 

 



5 Int. J. Anal. Appl. (2022), 20:23 

 

Table 2. Some numerical results of moments, variance, skewness, and kurtosis. 

Statistics  𝑏 = 1.5,𝑘 = 0.1 𝑏 = 1.7,𝑘 = 0.1 
Remarks 

𝜇/
𝑟 𝑎 = 0.1 𝑎 = 0.2 𝑎 = 0.01 𝑎 = 0.03 𝑎 = 0.05 

𝜇/
1
 0.4880 0.5063 0.4972 0.5009 0.5047 

D
e
c
re

a
si
n
g
 𝜇/

2
 0.3023 0.3208 0.3101 0.3138 0.3177 

𝜇/
3
 0.2155 0.2317 0.2214 0.2247 0.2281 

𝜇/
4
 0.1668 0.1807 0.1714 0.1743 0.1772 

𝜎2 0.0166 0.0046 0.0103 0.0079 0.0053 

𝛽1 0.0022 0.0006 0.0008 0.0005 0.0004 Decreasing 

𝛽2 0.0971 0.0645 0.0730 0.0671 0.0602 Decreasing 

 

Tables 1 and 2 illustrate decreasing behavior of the first four moments, variance, skewness, and 

kurtosis with some choices of model parameters. 

Moment generating function 𝑀𝑋(𝑡) can be defined as: 

𝑀𝑋(𝑡) = ∑
𝑡𝑟

𝑟!

∞

𝑟=0

𝜇𝑟
/
. 

 
Therefore, the moment generating function (mgf) of X is given by: 

𝑀𝑋−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑡) =
𝑎𝑏

(1 −𝑘𝑎)𝑏
 ∑

𝑡𝑟

𝑟!

∞

𝑟=0

∑ (
𝑏 −1
𝑗
)
(−1)𝑗𝑘𝑎𝑗

𝑟 +𝑎(𝑏 −𝑗)
(1−𝑘𝑟+𝑎(𝑏−𝑗)) 

∞

𝑗=0
. 

Characteristic function is defined as: 

∅𝑋(𝑡) = ∑
(𝑖𝑡)𝑟

𝑟!

∞

𝑟=0

𝜇
𝑟
′ . 

By following Equation (5), it is obtained as: 

∅𝑋−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑡) =
𝑎𝑏

(1−𝑘𝑎)𝑏
 ∑

(𝑖𝑡)𝑟

𝑟!

∞

𝑟=0

∑ (
𝑏 −1
𝑗
)
(−1)𝑗𝑘𝑎𝑗

𝑟 +𝑎(𝑏 −𝑗)
(1 −𝑘𝑟+𝑎(𝑏−𝑗)) 

∞

𝑗=0
. 

 

 

The factorial generating function of X is defined as: 

𝐹𝑥(𝑡) = 𝐸(1+𝑡)
𝑥 = 𝐸(𝑒𝑥𝑙𝑛(1+𝑡)) = ∑

(𝑙𝑛(1+𝑡))
𝑟

𝑟!

∞

𝑟=0

𝜇𝑟
′ . 

By using Equation (5), it is obtained as: 

𝐹𝑥−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑡) = (
𝑎𝑏

(1−𝑘𝑎)𝑏
 ∑

(𝑙𝑛(1+𝑡))
𝑟

𝑟!

∞

𝑟=0

∑
(
𝑏 −1
𝑗
)
(−1)𝑗𝑘𝑎𝑗

𝑟 +𝑎(𝑏 −𝑗)

(1−𝑘𝑟+𝑎(𝑏−𝑗)) 

∞

𝑗=0
).  



6 Int. J. Anal. Appl. (2022), 20:23 

 

Negative moments of X, substitute r with – w in Equation (5) and it is given by: 

𝜇−𝑤−𝐸𝐿𝑇𝑟−𝑃𝐹
′ =

𝑎𝑏

(1 −𝑘𝑎)𝑏
 ∑ (

𝑏 −1
𝑗
)
(−1)𝑗𝑘𝑎𝑗

𝑎(𝑏 −𝑗)−𝑤
(1−𝑘𝑎(𝑏−𝑗)−𝑤) 

∞

𝑗=0
.  

Furthermore, for fractional positive and fractional negative moments of X, substitute r with (
𝑚

𝑛
) and 

(−
𝑚

𝑛
) in Equation (6) respectively. In the theory of statistics, the Mellin transformation is famous 

as a distribution of the product as well as a quotient for independent random variables. The Mellin 

transformation is represented by 

𝑀𝑥(𝑚) = 𝐸(𝑥
𝑚−1) = ∫ 𝑥𝑚−1𝑓(𝑥)𝑑𝑥

𝑘

1

. 

Mellin transformation of X is given by: 

𝑀𝑥−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑚) =
𝑎𝑏

(1−𝑘𝑎)𝑏
 ∑ (

𝑏 −1
𝑗
)

(−1)𝑗𝑘𝑎𝑗

𝑎(𝑏 −𝑗)+𝑚 −1
(1−𝑘𝑎(𝑏−𝑗)+𝑚−1) 

∞

𝑗=0
. 

 
 

 

2.3. Incomplete moments  

The r – th lower incomplete moments of X is defined as: 

𝛷𝑟(𝑡) = ∫ 𝑥
𝑟𝑓(𝑥)𝑑𝑥

𝑡

𝑘
 , 

and it is given by: 

𝛷𝑟−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑡) =
𝑎𝑏

(1−𝑘𝑎)𝑏
 ∑ (

𝑏 −1
𝑗
)
(−1)𝑗𝑘𝑎𝑗

𝑟 +𝑎(𝑏 −𝑗)
(𝑡𝑟+𝑎(𝑏−𝑗) −𝑘𝑟+𝑎(𝑏−𝑗)) 

∞

𝑗=0
. (7) 

The first incomplete moment can be obtained by substituting r = 1 in Equation (7) as follows: 

𝛷1−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑡) =
𝑎𝑏

(1−𝑘𝑎)𝑏
 ∑ (

𝑏 −1
𝑗
)
(−1)𝑗𝑘𝑎𝑗

1 +𝑎(𝑏 −𝑗)
(𝑡1+𝑎(𝑏−𝑗) −𝑘1+𝑎(𝑏−𝑗)) 

∞

𝑗=0
. (8) 

 

The residual life function is the probability that a component whose life says x, will survive in an 

additional interval at t. It is given by: 

𝑅(𝑡 𝑥⁄ ) =
𝑆(𝑥 +𝑡)

𝑆(𝑡)
. 

Therefore, the residual life function of X is: 

𝑆𝑅(𝑡)−𝐸𝐿𝑇𝑟−𝑃𝐹(
𝑡
𝑥⁄ ) =

(1−𝑘𝑎)𝑏 −((𝑥 +𝑡)𝑎 −𝑘𝑎)𝑏

(1−𝑘𝑎)𝑏 −(𝑡𝑎 −𝑘𝑎)𝑏
, 𝑥 > 0. 

 

 

The reverse residual life is obtained by 𝑆�̅�(𝑡)−𝐸𝐿𝑇𝑟−𝑃𝑜𝑤(
𝑡
𝑥⁄ ) =

𝑆(𝑥−𝑡)

𝑆(𝑡)
 . The reverse residual life 

function of X is therefore given by: 

𝑆�̅�(𝑡)−𝐸𝐿𝑇𝑟−𝑃𝐹(
𝑡
𝑥⁄ ) =

(1−𝑘𝑎)𝑏 −((𝑥 −𝑡)𝑎 −𝑘𝑎)𝑏

(1−𝑘𝑎)𝑏 −(𝑡𝑎 −𝑘𝑎)𝑏
, 𝑥 > 0.  

Mean residual life (MRL) function is defined as 
1−𝛷1(𝑡)

𝑆(𝑡)−𝑡
. It is obtained for X as 



7 Int. J. Anal. Appl. (2022), 20:23 

 

MRL =
1−

𝑎𝑏

(1−𝑘𝑎)𝑏
 ∑ (

𝑏 −1
𝑗
)
(−1)𝑗𝑘𝑎𝑗

1+𝑎(𝑏−𝑗)
(𝑡1+𝑎(𝑏−𝑗) −𝑘1+𝑎(𝑏−𝑗)) ∞𝑗=0

𝑆(𝑡) −𝑡
. 

Mean inactivity time (MIT) is defined as  𝑡 −
𝛷1(𝑡)

𝑃(𝑡)
. It is obtained for X as 

MRL = 𝑡 −
1

𝑃(𝑡)
(

𝑎𝑏

(1−𝑘𝑎)𝑏
 ∑ (

𝑏 −1
𝑗
)
(−1)𝑗𝑘𝑎𝑗

1+𝑎(𝑏 −𝑗)
(𝑡1+𝑎(𝑏−𝑗) −𝑘1+𝑎(𝑏−𝑗)) 

∞

𝑗=0
) 

Vitality function is defined as 𝑉(𝑥) =
1

𝑆(𝑥)
∫ 𝑥𝑓(𝑥)𝑑𝑥
1

𝑥
. It is obtained for X as 

𝑉(𝑥) =
1

1−𝐹(𝑥)
(

𝑎𝑏

(1−𝑘𝑎)𝑏
 ∑ (

𝑏 −1
𝑗
)
(−1)𝑗𝑘𝑎𝑗

1+𝑎(𝑏 −𝑗)
(1−𝑥1+𝑎(𝑏−𝑗)) 

∞

𝑗=0
). 

The conditional moments are defined as𝐸(𝑥𝑟|𝑥 > 𝑡) =
1

�̅�(𝑡)
∫ 𝑥𝑟𝑓(𝑥)𝑑𝑥
1

𝑡
. It is obtained for X as 

𝐸(𝑥𝑟|𝑥 > 𝑡) =
1

1−𝑃(𝑡)
(

𝑎𝑏

(1−𝑘𝑎)𝑏
 ∑ (

𝑏 −1
𝑗
)
(−1)𝑗𝑘𝑎𝑗

𝑟 +𝑎(𝑏 −𝑗)
(1−𝑡𝑟+𝑎(𝑏−𝑗)) 

∞

𝑗=0
) 

 

2.4. Bonferroni and Lorenz curves 

The Bonferroni 𝐵(𝑥) and Lorenz 𝐿(𝑥) curves are important not only in the study of economics, the 

distribution of income, poverty, or wealth, but they play a vital role in the fields of insurance, 

demography, medicine, reliability, and others. These curves are defined respectively by: 

𝐵(𝑥) =
∫ 𝑥𝑓(𝑥)𝑑𝑥
𝑡

0

𝜇1
/ ,  𝐿(𝑥) =

𝐵(𝑥)

𝐹(𝑥)
, 

Lorenz curve 𝐿(𝑥) 

𝐿𝐸𝐿𝑇𝑟−𝑃𝐹(𝑡) =
∑ (

𝑏 −1
𝑗
)
(−1)𝑗𝑘𝑎𝑗

1+𝑎(𝑏−𝑗)
(𝑡1+𝑎(𝑏−𝑗) −𝑘1+𝑎(𝑏−𝑗)) ∞𝑗=0

∑ (
𝑏 −1
𝑗
)
(−1)𝑗𝑘𝑎𝑗

1+𝑎(𝑏−𝑗)
(1−𝑘1+𝑎(𝑏−𝑗)) ∞𝑗=0

, (9) 

and Bonferroni curve 𝐵(𝑥) are given by: 

𝐵𝐸𝐿𝑇𝑟−𝑃𝐹(𝑡) =
∑ (

𝑏 −1
𝑗
)
(−1)𝑗𝑘𝑎𝑗

1+𝑎(𝑏−𝑗)
(𝑡1+𝑎(𝑏−𝑗) −𝑘1+𝑎(𝑏−𝑗)) ∞𝑗=0

((
𝑥𝑎−𝑘𝑎

1−𝑘𝑎
)
𝑏
)∑ (

𝑏 −1
𝑗
)
(−1)𝑗𝑘𝑎𝑗

1+𝑎(𝑏−𝑗)
(1−𝑘1+𝑎(𝑏−𝑗)) ∞𝑗=0

. 

 

2.5.Reliability measures 

The survival function is defined as the probability that a component will survive till 

time x. Analytically, it is defined as: 

𝑆(𝑥) = 1−𝐹(𝑥). 

The survival function of X is therefore given by: 



8 Int. J. Anal. Appl. (2022), 20:23 

 

𝑆𝐸𝐿𝑇𝑟−𝑃𝐹(𝑥;𝑎,𝑏) = 1−(
𝑥𝑎 −𝑘𝑎

1−𝑘𝑎
)

𝑏

. 

 

 

The hazard rate function (HRF) is defined as measuring the failure rate of a component in a particular 

time x. Mathematically, it is defined as: 

ℎ(𝑥) = 𝑓(𝑥) 𝑆(𝑥)⁄ . 

Hence, the hazard rate function of X is given by: 

ℎ𝐸𝐿𝑇𝑟−𝑃𝐹(𝑥;𝑎,𝑏) =
𝑎𝑏𝑥𝑎−1(𝑥𝑎 −𝑘𝑎)𝑏−1 

(1 −𝑘𝑎)𝑏 −(𝑥𝑎 −𝑘𝑎)𝑏
.  

 

 

Further, several notable reliability measures may be derived for X such as the reversed hazard rate 

function. It is defined as: 

ℎ𝑟(𝑥) = 𝑓(𝑥) 𝐹(𝑥)⁄ . 

The reversed hazard rate function of X is given by: 

ℎ𝑟−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑥;𝑎,𝑏) =
𝑎𝑏𝑥𝑎−1

(𝑥𝑎 −𝑘𝑎)
. 

 

 

The Mills ratio is defined as 𝑀(𝑥) = 𝑆(𝑥) 𝑓(𝑥)⁄ . Hence, the Mills ratio of X is given by: 

𝑀𝐸𝐿𝑇𝑟−𝑃𝐹(𝑥;𝑎,𝑏) =
(1−𝑘𝑎)𝑏 −(𝑥𝑎 −𝑘𝑎)𝑏

𝑎𝑏𝑥𝑎−1(𝑥𝑎 −𝑘𝑎)𝑏−1 
 . 

 

 

The Odd function is defined as 𝑂(𝑥) = 𝐹(𝑥) 𝑆(𝑥)⁄ . Therefore, the Odd function of X is given by: 

𝑂𝐸𝐿𝑇𝑟−𝑃𝐹(𝑥;𝑎,𝑏) = (
𝑥𝑎 −𝑘𝑎

1−𝑘𝑎
)

−𝑏

−1.  

 

3. Miscellaneous measures 

This section covers several measures including limiting behavior, shapes of density and hazard rate 

functions, quantile function, entropy measures, and distribution of order statistics, bivariate, and 

multivariate extensions for ELTr-PF distribution. 

 

3.1.Limiting behavior 

The limiting behavior of the CDF, PDF, and HRF of X for x → 𝑘 and x → 1 is discussed in propositions 

1 and 2. 

Proposition 1. Limiting behaviors of the CDF, PDF, and HRF of X for x → 𝑘 are given respectively 

by: 

 𝐹𝐸𝐿𝑇𝑟−𝑃𝐹 (𝑘)~0,  

 𝑓𝐸𝐿𝑇𝑟−𝑃𝐹(𝑘)~0,  

 ℎ𝐸𝐿𝑇𝑟−𝑃𝐹 (𝑘)~0. 
 

 



9 Int. J. Anal. Appl. (2022), 20:23 

 

Proposition 2. Limiting behaviors of the CDF, PDF, and HRF of X for x → 1 are given respectively 

by: 

 𝐹𝐸𝐿𝑇𝑟−𝑃𝐹 (1)~1,  

 𝑓𝐸𝐿𝑇𝑟−𝑃𝐹 (1)~
𝑎𝑏

(1−𝑘𝑎) 
,  

 
ℎ𝐸𝐿𝑇𝑟−𝑃𝐹 (1)~𝐼𝑛𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑒. 

 
 

 

3.2.Shapes of density and hazard rate functions  

Different curves of PDF and HRF of X are presented in Figures 1 and 2, for different choices of 

model parameters. Note that in Figure 1, curves of the PDF present some possible shapes including 

increasing, upside-down increasing, and decreasing. However, possible shapes of the HRF in Figure 

2 present increasing and bathtub-shaped. 

 

(a) 

 

(b) 

Figure 1. Plots of PDF (a) and HRF (b) for ELTr-PF distribution. 

 

3.3.Quantile function 

The concept of quantile function was introduced by [18]. The qth quantile function of the ELTr-PF 

distribution is obtained by inverting the CDF in Equation (1). It is defined by: 

𝑞 = 𝐹(𝑥𝑞) = 𝑃(𝑋 ≤ 𝑥𝑞), 𝑞 ∈ (0,1). 

Then, the quantile function of X is given by: 

𝑥𝑞−𝐸𝐿𝑇𝑟−𝑃𝐹 = (𝑘
𝑎 +(1−𝑘𝑎)𝑞

1
𝛽⁄ )

1
𝛼⁄

. (10) 

To derive the 1st quartile, median and 3rd quartile of X, one may place q = 0.25, 0.5, and 0.75 

respectively in Equation (10). Henceforth, to generate random numbers, one may assume that the 

expression in Equation (10) follows to uniform distribution u= U (0, 1). 



10 Int. J. Anal. Appl. (2022), 20:23 

 

3.4.Entropy measures 

This subsection covers several well-known entropy measures addressed by ([19], [20], [21], [22], [23], 

[24]).  

The entropy of r.v. X is a measure of uncertainty. The Rényi entropy of X is defined by: 

𝐼𝛿(𝑋) =
1

1 −𝛿
𝑙𝑜g∫𝑓𝛿(𝑥)𝑑𝑥

1

𝑘

 , 𝛿 > 0 𝑎𝑛𝑑𝛿 ≠ 1. (11) 

First,  𝑓𝐸𝐿𝑇𝑟−𝑃𝑜𝑤(𝑥) is simplified in terms of 𝑓
𝛿
𝐸𝐿𝑇𝑟−𝑃𝑜𝑤

(𝑥) by considering Equation (2) as: 

𝑓𝛿
𝐸𝐿𝑇𝑟−𝑃𝐹 

(𝑥) = (
𝑎𝑏

(1−𝑘𝑎)𝑏
 )
𝛿

𝑥𝛿(𝑎−1)(𝑥𝑎 −𝑘𝑎)𝛿(𝑏−1 ), 

by applying the binomial expansion, 

𝑓𝛿
𝐸𝐿𝑇𝑟−𝑃𝐹 

(𝑥) = (
𝑎𝑏

(1−𝑘𝑎)𝑏
 )
𝛿

∑(
𝛿(𝑏 −1)

𝑖
)

∞

𝑖=0

(−1)𝑖𝑘𝑎𝑖𝑥𝛿(𝑎𝑏−1)−𝑎𝑖, 

 
and substituting this into Equation (11) gives the Rényi entropy of X as: 

𝐼𝛿−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑋) = (
𝑎𝑏

(1−𝑘𝑎)𝑏
 )
𝛿

∑(
𝛿(𝑏 −1)

𝑖
)

∞

𝑖=0

(−1)𝑖𝑘𝑎𝑖 ∫𝑥𝛿(𝑎𝑏−1)−𝑎𝑖𝑑𝑥

1

𝑘

 ,   

hence, by integrating the last expression the reduced form of the Rényi entropy for X is obtained and 

it is given by: 

𝐼𝛿−𝐸𝐿𝑇𝑟−𝑃𝐹 (𝑋) = (
𝑎𝑏

(1−𝑘𝑎)𝑏
 )
𝛿

𝑙𝑜g∑𝐴𝑖,𝛿
1

𝜓𝑖,𝛿
(1−𝑘𝜓𝑖,𝛿)

∞

𝑖=0

, (12) 

where 𝜓𝑖,𝛿 = 𝛿(𝑎𝑏 −1)−𝑎𝑖, 𝐴𝑖,𝛿 = (
𝛿(𝑏 −1)

𝑖
)(−1)𝑖𝑘𝑎𝑖. 

A generalization of the Boltzmann-Gibbs entropy is the  – entropy. Although in physics, it is referred 

to as the Tsallis entropy. Tsallis entropy /  – entropy is defined by 

𝐻 (𝑋) =
1

−1
(1− ∫𝑓 (𝑥)𝑑𝑥

1

𝑘

) , > 0 𝑎𝑛𝑑 ≠ 1. 

The Tsallis entropy of X is given by 

𝐻 −𝐸𝐿𝑇𝑟−𝑃𝐹(𝑋) =
1

−1
(1−(

𝑎𝑏

(1−𝑘𝑎)𝑏
 ) 𝑙𝑜g∑𝐴𝑖,

1

𝜓𝑖,
(1−𝑘𝜓𝑖, )

∞

𝑖=0

), 

 

 

where 𝜓𝑖, = (𝑎𝑏 −1)−𝑎𝑖, 𝐴𝑖, = (
(𝑏 −1)
𝑖

)(−1)𝑖𝑘𝑎𝑖. 

The Havrda and Charvat introduced 𝜔 − entropy measure. It is defined by 

𝐻𝜔(𝑋) =
1

21−𝜔 −1
(∫𝑓𝜔(𝑥)𝑑𝑥

1

𝑘

−1) , 𝜔 > 0 𝑎𝑛𝑑 𝜔 ≠ 1. 



11 Int. J. Anal. Appl. (2022), 20:23 

 

Havrda and Charvat entropy of X is given by 

𝐻𝜔−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑋) =
1

21−𝜔 −1
(((

𝑎𝑏

(1−𝑘𝑎)𝑏
 )
𝜔

𝑙𝑜g∑𝐴𝑖,𝜔
1

𝜓𝑖,𝜔
(1−𝑘𝜓𝑖,𝜔)

∞

𝑖=0

)−1), 

 

 

where 𝜓𝑖,𝜔 = 𝜔(𝑎𝑏 −1)−𝑎𝑖, 𝐴𝑖,𝜔 = (
𝜔(𝑏 −1)

𝑖
)(−1)𝑖𝑘𝑎𝑖. 

Arimoto generalized the work of Havrda and Charvat by introducing − entropy measure. It is 

defined by 

𝐻 (𝑋) =
21− −1

(

 (∫𝑓 (𝑥)𝑑𝑥

1

𝑘

)

1

−1

)

  , > 0 𝑎𝑛𝑑 ≠ 1. 

 Arimoto entropy of X is given by 

𝐻 −𝐸𝐿𝑇𝑟−𝑃𝐹(𝑋) =
21− −1

(

 ((
𝑎𝑏

(1−𝑘𝑎)𝑏
 )

1
 

𝑙𝑜g∑𝐴
𝑖,
1

1

𝜓
𝑖,
1
(1−𝑘

𝜓
𝑖,
1

)

∞

𝑖=0

)

−1 )

 ,  

where 𝜓
𝑖,
1
 
=
1
(𝑎𝑏 −1)−𝑎𝑖, 𝐴

𝑖,
1
 
= (

1
(𝑏 −1)

𝑖
)(−1)𝑖𝑘𝑎𝑖. 

Booker and Lubba developed the 𝜏 − entropy measure. It is defined by  

𝐻𝜏(𝑋) =
𝜏

𝜏 −1

(

 1−(∫𝑓𝜏 (𝑥)𝑑𝑥

1

𝑘

)

1

𝜏

)

  , 𝜏 > 0 𝑎𝑛𝑑 𝜏 ≠ 1. 

Boekee and Lubba entropy of X is given by 

𝐻𝜏−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑋) =
𝜏

𝜏 −1
(1−((

𝑎𝑏

(1−𝑘𝑎)𝑏
 )
𝜏

𝑙𝑜g∑𝐴𝑖,𝜏
1

𝜓𝑖,𝜏
(1−𝑘𝜓𝑖,𝜏)

∞

𝑖=0

)

1

𝜏 

). 

 

 

where 𝜓𝑖,𝜏 = 𝜏(𝑎𝑏 −1)−𝑎𝑖, 𝐴𝑖,𝜏 = (
𝜏(𝑏 −1)

𝑖
)(−1)𝑖𝑘𝑎𝑖. 

Mathai and Haubold generalized the classical Shannon entropy is known as − entropy. It is defined 

by 

𝐻 (𝑋) =
1

−1
(∫𝑓2− (𝑥)𝑑𝑥

1

𝑘

−1) , > 0 𝑎𝑛𝑑 ≠ 1. 

Mathai and Haubold entropy of X is given by 

𝐻 −𝐸𝐿𝑇𝑟−𝑃𝐹(𝑋) =
1

−1
(((

𝑎𝑏

(1−𝑘𝑎)𝑏
 )
2−

𝑙𝑜g∑𝐴𝑖,2−
1

𝜓𝑖,2−
(1 −𝑘𝜓𝑖,2− )

∞

𝑖=0

)−1), 

 

 



12 Int. J. Anal. Appl. (2022), 20:23 

 

where 𝜓𝑖,2− = (2− )(𝑎𝑏 −1)−𝑎𝑖, 𝐴𝑖,2− = (
(2− )(𝑏 −1)

𝑖
)(−1)𝑖𝑘𝑎𝑖. 

 

Table 3 presents the flexible behavior of the entropy measures for some choices of model parameters 

for S-I (𝒂 = 𝟏.𝟏,𝒃 = 𝟐.𝟏,𝒌 = 𝟎.𝟎𝟏), S-II(𝒂 = 𝟏.𝟏,𝒃 = 𝟏.𝟓,𝒌 = 𝟎.𝟎𝟏𝟓), and S-III(𝒂 = 𝟏.𝟓,𝒃 =

𝟏.𝟏,𝒌 = 𝟎.𝟎𝟓).  

 

Table 3. Some numerical results of Rényi, Tsallis, Havrda and Charvat, Arimoto, Boekee and Lubba, 

Mathai and Haubold entropy measures. 

Entropy Int. S-I S-II S-III 

Rényi 

 

𝛿 = 1.1 4.8902 1.3591 0.4750 

𝛿 = 1.5 1.3337 0.3706 0.1295 

𝛿 = 1.7 1.0796 0.3000 0.1048 

𝛿 = 1.9 0.9385 0.2608 0.0911 

Tsallis 

 

= 1.1 0.3634 0.1047 0.0323 

= 1.5 0.1500 0.0431 -0.0070 

= 1.7 0.0780 0.0181 -0.0248 

= 1.9 0.0174 -0.0044 -0.0417 

Havrda and 

Charvat 

𝜔 = 1.1 0.0028 0.0019 0.0045 

𝜔 = 1.5 0.0820 0.0389 0.0554 

𝜔 = 1.7 0.1887 0.0851 0.1109 

𝜔 = 1.9 0.3680 0.1601 0.1969 

Arimoto 

 

= 1.1 0.0039 0.0010 0.0004 

= 1.5 0.3128 0.0529 0.0213 

= 1.7 1.5767 0.1481 0.0596 

= 1.9 85.315 0.3558 0.1333 

Boekee and Lubba 

 

𝜏 = 1.1 0.3701 0.1063 0.0333 

𝜏 = 1.5 0.2238 0.0665 0.0087 

𝜏 = 1.7 0.1871 0.0552 0.0012 

𝜏 = 1.9 0.1606 0.0466 -0.0045 

Mathai and 

Haubold 

 

= 1.1 -0.1569 0.0401 0.3761 

= 1.5 -0.1500 -0.0431 0.0070 

= 1.7 -0.1031 -0.0306 -0.0051 

= 1.9 -0.0403 -0.0116 -0.0035 



13 Int. J. Anal. Appl. (2022), 20:23 

 

Table 3 presents’ versatile behavior of entropy measures for different parametric values. Note that 

the Rényi entropy is decreasing, Tsallis entropy is decreasing, Havrda and Charvat entropy is 

increasing, Arimoto entropy is increasing, Boekee and Lubba entropy is decreasing, and Mathai and 

Haubold's entropy is decreasing. 

 

3.5.Distribution of order statistics 

This subsection covers i-th order statistics PDF, minimum order statistics PDF, maximum order 

statistics PDF, order statistics CDF, median order statistics PDF, and Joint order statistics PDF.   

In reliability analysis and life testing of a component in quality control, OS has a noteworthy 

contribution. Let X1 , X2 , X3 , ..., Xn be a random sample of size n which follows the ELTr-PF 

distribution and {X(1) < X(2) <X(3) < ...<X(n) }be the corresponding order statistics. The PDF of the 

i-th OS is given by: 

𝑓(𝑖:𝑛)(𝑥) =
1

𝐵(𝑖,𝑛−𝑖+1)!
(𝐹(𝑥))

𝑖−1
(1−𝐹(𝑥))

𝑛−𝑖
𝑓(𝑥),  i=1, 2, 3,…, n. 

 

By incorporating Equations (3) and (4), the PDF of i-th OS is given by: 

𝑓(𝑖:𝑛)−𝐸𝐿𝑇𝑟−𝑃𝐹 (𝑥) = (

1

𝐵(𝑖,𝑛−𝑖+1)!
((
𝑥𝑎−𝑘𝑎

1−𝑘𝑎
)
𝑏

)
𝑖−1

(1−(
𝑥𝑎−𝑘𝑎

1−𝑘𝑎
)
𝑏

)
𝑛−𝑖

×

(
𝑎𝑏

(1−𝑘𝑎)𝑏
 𝑥𝑎−1(𝑥𝑎 −𝑘𝑎)𝑏−1 )

). (13) 

For minimum OS, substitute (i = 1) into Equation (13) as; 

𝑓(1:𝑛)−𝐸𝐿𝑇𝑟−𝑃𝐹 (𝑥) =

(

 
 

1

𝐵(𝑖,𝑛 −𝑖 +1)!
(1−(

𝑥𝑎 −𝑘𝑎

1−𝑘𝑎
)

𝑏

)

𝑛−1

×

(
𝑎𝑏

(1−𝑘𝑎)𝑏
 𝑥
𝑎−1(𝑥𝑎 −𝑘𝑎)𝑏−1 )

)

 
 
.  

While the maximum OS is obtained by substituting (i = n) into Equation (13) as: 

𝑓(𝑛:𝑛)−𝐸𝐿𝑇𝑟−𝑝𝐹 (𝑥) =
1

𝐵(𝑖,𝑛 −𝑖 +1)!
((
𝑥𝑎 −𝑘𝑎

1−𝑘𝑎
)

𝑏

)

𝑛−1

(
𝑎𝑏

(1−𝑘𝑎)𝑏
 𝑥
𝑎−1(𝑥𝑎 −𝑘𝑎)𝑏−1 ). 

 

 

Correspondingly, the CDF of the i-th OS is defined by: 

𝐹(𝑖:𝑛)(𝑥) = ∑(
𝑛
𝑟
)

𝑛

𝑟=𝑖

(𝐹(𝑥))
𝑟
(1−𝐹(𝑥))

𝑛−𝑟
. 

By incorporating Equation (1), the CDF of the i-th OS is obtained and it is given by: 



14 Int. J. Anal. Appl. (2022), 20:23 

 

𝐹(𝑖:𝑛)−𝐸𝐿𝑇𝑟−𝑃𝐹 (𝑥) = ∑(
𝑛
𝑟
)

𝑛

𝑟=1

(((
𝑥𝑎 −𝑘𝑎

1−𝑘𝑎
)

𝑏

)

𝑛−1

)

𝑟

(1−((
𝑥𝑎 −𝑘𝑎

1−𝑘𝑎
)

𝑏

)

𝑛−1

)

𝑛−𝑟

. 

The Median of the i-th OS is defined by: 

𝑓(𝑚+1:𝑛)(𝑥) =
(2𝑚 +1)!

(𝑚!)2
𝑓(𝑥)(𝐹(𝑥))

𝑚
(1−𝐹(𝑥))

𝑚
. 

 

By incorporating Equations (1) and (2), the PDF of the median Xm+1 OS is obtained as: 

𝑓(𝑚+1:𝑛)−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑥) =
(2𝑚+1)!

(𝑚!)2
(((

𝑥𝑎 −𝑘𝑎

1−𝑘𝑎
)

𝑏

)

𝑛−1

)

𝑚

((1−(
𝑥𝑎 −𝑘𝑎

1−𝑘𝑎
)

𝑏

)

𝑛−1

)

𝑚

. 

The Joint distribution of the i-th and j-th OS is defined by: 

𝑓(𝑖:𝑗:𝑛)(𝑥𝑖,𝑥𝑗) = 𝐶(𝐹(𝑥𝑖))
𝑖−1
(𝐹(𝑥𝑗)−𝐹(𝑥𝑖))

𝑗−𝑖−1
(1−𝐹(𝑥𝑗))

𝑛−𝑗
𝑓(𝑥𝑖)𝑓(𝑥𝑗). (14) 

To obtain the joint distributions, Equations (1) and (2) are further written as follows: 

𝐹𝐸𝐿𝑇𝑟−𝑃𝐹 (𝑥𝑖) = (
𝑥𝑖
𝑎 −𝑘𝑎

1−𝑘𝑎
)

𝑏

,𝐹𝐸𝐿𝑇𝑟−𝑃𝑜𝑤 (𝑥𝑗) = (
𝑥𝑗
𝑎 −𝑘𝑎

1−𝑘𝑎
)

𝑏

, 

𝑓𝐸𝐿𝑇𝑟−𝑃𝐹 (𝑥𝑖) =
𝑎𝑏

(1−𝑘𝑎)𝑏
 𝑥𝑖

𝑎−1(𝑥𝑖
𝑎 −𝑘𝑎)𝑏−1 , 

𝑓𝐸𝐿𝑇𝑟−𝑃𝐹 (𝑥𝑗) =
𝑎𝑏

(1−𝑘𝑎)𝑏
 𝑥𝑗

𝑎−1(𝑥𝑗
𝑎 −𝑘𝑎)

𝑏−1 
. 

By substituting these expressions into Equation (14), the joint distribution of the i-th and j-th OS is 

obtained as: 

𝑓(𝑖:𝑗:𝑛)−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑥𝑖,𝑥𝑗) =

(

 
 
 
 
 
 
 
 
 𝐶((

𝑥𝑖
𝑎 −𝑘𝑎

1−𝑘𝑎
)

𝑏

)

𝑖−1

((
𝑥𝑗
𝑎 −𝑘𝑎

1−𝑘𝑎
)

𝑏

−(
𝑥𝑖
𝑎 −𝑘𝑎

1−𝑘𝑎
)

𝑏

)

𝑗−𝑖−1

×

(1−(
𝑥𝑗
𝑎 −𝑘𝑎

1−𝑘𝑎
)

𝑏

)

𝑛−𝑗

×

((
𝑎𝑏

(1−𝑘𝑎)𝑏
 )
2

𝑥𝑖
𝑎−1(𝑥𝑖

𝑎 −𝑘𝑎)𝑏−1 )×

(𝑥𝑗
𝑎−1(𝑥𝑗

𝑎 −𝑘𝑎)
𝑏−1 

)

)

 
 
 
 
 
 
 
 
 

. 

 

3.6.Bivariate and multivariate extensions 

In this subsection, we discuss the bivariate and multivariate extensions for the ELTr–PF distribution 

by following the Morgenstern and the Clayton families.  



15 Int. J. Anal. Appl. (2022), 20:23 

 

The CDF of the Bi–ELTr–PF distribution followed by the Morgenstern family for the random vector 

(𝑉1,𝑉2) is  

𝐹𝜙−𝐸𝐿𝑇𝑟−𝑃𝐹 (𝑉1,𝑉2) = (1+𝜙(1−𝐹1(𝑣1))(1−𝐹2(𝑣2)))𝐹1(𝑣1)𝐹2(𝑣2), 

where |𝜙| ≤ 1, 𝐹1(𝑣1) = (
𝑣1
𝑎−𝑘𝑎

1−𝑘𝑎
)
𝑏

, and 𝐹2(𝑣2) = (
𝑣2
𝑎−𝑘𝑎

1−𝑘𝑎
)
𝑏

. 

The CDF of the Bi– ELTr–PF distribution followed by the Clayton family for the random vector 

(𝑋,𝑌) is  

𝐶(𝑥,𝑦) = (𝑥−( 1+ 2) +𝑦−( 1+ 2) −1)
− 

1

( 1+ 2) ; 1 + 2 ≥ 0. 

Let 𝑣1~ ELTr–PF (𝛼1,𝛽1), and 𝑣2~ ELTr–PF (𝛼2,𝛽2). Then setting 𝑥 = 𝐹1(𝑣1) = (
𝑣1
𝑎−𝑘𝑎

1−𝑘𝑎
)
𝑏

and 

𝑦 = 𝐹2(𝑣2) = (
𝑣2
𝑎−𝑘𝑎

1−𝑘𝑎
)
𝑏

. 

The CDF of the Bi– ELTr–PF distribution followed by the Clayton family for the random vector 

(𝑉1,𝑉2) is  

𝐺𝐵𝑖−𝐸𝐿𝑇𝑟–𝑃𝐹 (𝑣1,𝑣2) = ((
𝑣1
𝑎 −𝑘𝑎

1−𝑘𝑎
)

−𝑏( 1+ 2)

+(
𝑣2
𝑎 −𝑘𝑎

1−𝑘𝑎
)

−𝑏( 1+ 2)

−1)

− 
1

( 1+ 2)

. 

A simple n-dimensional extension of the last version will be 

𝐻𝑀𝑢𝑙𝑡𝑖−𝐸𝐿𝑇𝑟–𝑃𝐹 (𝑥1,𝑥2,𝑥3,…,𝑥𝑛) = (∑((
𝑥𝑖
𝑎𝑖 −𝑘𝑖

𝑎𝑖

1−𝑘𝑖
𝑎𝑖
)

−𝑏( 1+ 2)

)

𝑛

𝑖=1

+1−𝑛)

− 
1

( 1+ 2)

. 

 

4. Inference 

In this section, the X's parameters are estimated by following the method of maximum 

likelihood estimation (MLE), as this method provides full information about the unknown model 

parameter. 

Let 𝑋1,𝑋2, . . . ,𝑋𝑛 be a random sample of size n from X, then the likelihood function 𝐿(𝜙) =

∏ 𝑓(𝑥𝑖;𝑎,𝑏)
𝑛
𝑖=1  of X is given by: 

𝐿𝐸𝐿𝑇𝑟−𝑃𝐹(𝜙) = (
𝑎𝑏

(1−𝑘𝑎)𝑏
 )
𝑛

∏𝑥𝑖
𝑎−1(𝑥𝑖

𝑎 −𝑘𝑎)𝑏−1 .

𝑛

𝑖=1

 

The log-likelihood function, 𝑙(𝜙) is thus given by: 



16 Int. J. Anal. Appl. (2022), 20:23 

 

𝑙𝐸𝐿𝑇𝑟−𝑃𝐹(𝜙) =

(

 
 
 
𝑛𝑙𝑜g𝑎 +𝑛𝑙𝑜g𝑏 −𝑛𝑙𝑜g(1−𝑘𝑎)+(𝑎 −1)∑𝑙𝑜g𝑥𝑖

𝑛

𝑖=1

+(𝑏 −1)∑𝑙𝑜g(𝑥𝑖
𝑎 −𝑘𝑎)

𝑛

𝑖=1 )

 
 
 
. (15) 

Partial derivatives for parameters 𝑎 and 𝑏 of Equation (15) yield the following: 

𝜕𝑙𝐸𝐿𝑇𝑟−𝑃𝐹(𝜙)

𝜕𝑎
=
𝑛

𝑎
+
𝑛𝑘𝑎𝑙𝑜g𝑎

1 −𝑘𝑎
+∑𝑙𝑜g𝑥𝑖

𝑛

𝑖=1

+(𝑏 −1)∑
𝑥𝑖
𝑎𝑙𝑜g𝑥𝑖 −𝑘

𝑎𝑙𝑜g𝑘

(𝑥𝑖
𝑎 −𝑘𝑎)

𝑛

𝑖=1

,  

𝜕𝑙𝐸𝐿𝑇𝑟−𝑃𝐹(𝜙)

𝜕𝑏
=
𝑛

𝑏
+∑𝑙𝑜g(𝑥𝑖

𝑎 −𝑘𝑎)

𝑛

𝑖=1

.  

The maximum likelihood estimates (�̂�𝑖 = 𝑎,̂ �̂�) of the ELTr-PF distribution can be obtained 

by maximizing Equation (15) or by solving the above non-linear equations simultaneously. These 

non-linear equations although do not provide an analytical solution for the MLEs and the optimum 

value of 𝑎, and 𝑏. Consequently, the Newton-Raphson type algorithm is an appropriate choice in the 

support of MLEs. 

 

5. Simulation 

In this subsection, we discuss the following algorithm (step 1 to 5) to observe the asymptotic 

performance of MLE’s �̂� = (�̂�, �̂�). 

Step -1: A random sample x1, x2, x3, ..., xn of sizes n = 100, 200, 300, 400, and 1000 from 

Equation (14). 

Step -2: Results of root mean square error (RMSE), coverage probability (CP), and average 

width (AW) are calculated with the assist of statistical software R. These results are presented in 

Tables 4 and 5.  

Step -3: Each sample is replicated 1000 times.  

Step -4: Gradual decrease with the increase in sample sizes is observed for RMSEs. 

Step -5: CPs of all the parameters 𝜙 = (𝛼,𝛽) is approximately 0.975 approaches to the 

nominal value and AW decreases as sample sizes increase.   

Furthermore, the following measures are defined in the development of average estimate 

(AE), RMSE, CP, and AW, where I(.) is an indicator function and 𝑠𝑒�̂�𝑖 =
√𝑣𝑎𝑟�̂�𝑖 is the standard 

error of estimate 𝜙𝑖.and it is given as follows: 



17 Int. J. Anal. Appl. (2022), 20:23 

 

𝐴𝐸(�̂�) =
1

𝑁
∑�̂�

𝑁

𝑖=1

,𝑅𝑀𝑆𝐸(�̂�) = √
1

𝑁
∑(�̂�𝑖 −𝜙)

2
𝑁

𝑖=1

, 

𝐶𝑃(�̂�) = ∑𝐼(�̂�𝑖 −0.95𝑠𝑒�̂�𝑖, �̂�𝑖 +0.95𝑠𝑒�̂�𝑖)

𝑁

𝑖=1

,and 

𝐴𝑊(�̂�) =
1

𝑁
∑|(�̂�𝑖 +0.95)−(�̂�𝑖 −0.95)|

𝑁

𝑖=1

 

 

Table 4. Root mean square error, coverage probability, and average width for (𝑎 =1.1, 𝑏 =1.1, 

k=0.01). 

Sample RMSE(a) CP(a) AW(a) 

100 0.5848 1 3.4874 

200 0.5359 1 2.1784 

300 0.5078 1 2.2354 

400 0.4991 1 2.0208 

1000 0.3330 1 1.1878 

 

Table 5. Root mean square error, coverage probability, and average width for (𝑎 =1.1, 𝑏 =1.1, 

k=0.01). 

Sample RMSE(b) CP(b) AW(b) 

100 0.2320 0.98 1.9249 

200 0.2339 0.98 1.4396 

300 0.2259 0.98 1.1951 

400 0.2235 0.98 1.0603 

1000 0.2020 0.98 0.6999 

 

In both Tables 4 and 5, the RMSE and AW values reduced as the sample size increases. This 

indicates that the parameters of the ELTr-PF distribution are good and stable. 

 

 

 



18 Int. J. Anal. Appl. (2022), 20:23 

 

6. Analysis of Real-Life Data 

In this section, the application of the ELTr-PF distribution is discussed. For this, two 

engineering datasets are explored. The ELTr-PF distribution is compared with its competing models 

(presented in Table 6) based on some criteria called, -Log-likelihood (-LL), Akaike information 

criterion (AIC), Bayesian information criterion (BIC), along with the good-of-fit statistics Cramer-

Von Mises (CM), Anderson-Darling (AD), and Kolmogorov Smirnov (K-S) with its p-value. All the 

numerical results are calculated with the assistance of statistical software R with its exclusive package 

AdequacyModel (https://www.r-project.org/).  

 

Table 6. List of Some Competitive models CDFs. 

Model Model’s CDFs 
Parameter / 

variable Range 
Author(s) 

L-I 𝐺𝐼(𝑥) = 𝑥
𝑎  

𝑎 > 0, 0 < x < 1 [25] 
L-II 𝐺𝐼𝐼(𝑥) = 1−(1−𝑥)

𝑎  

Beta 𝐺𝐼𝐼𝐼(𝑥) = 𝐼𝑥(𝑎,𝑏) 
𝑎,𝑏 > 0  

0 < x < 1 
 

Topp-Leone 𝐺𝐼𝑉(𝑥) = (2𝑥 −𝑥
2)𝑎 𝑎 > 0, 0 < x < 1 [11] 

Kum 𝐺𝑉(𝑥) = 1 −(1 −𝑥
𝑎 )𝑏 

𝑎,𝑏 > 0,  

0 < x < 1 
[10] 

GPF 𝐺𝑉𝐼(𝑥) = 1−(g−𝑥)
𝑎(g−𝑘)−𝑎 

𝑎 > 0 

𝑘 < 𝑥 <  g 
[26] 

WPF 
𝐺𝑉𝐼𝐼(𝑥) = 1− 𝑒

−𝑎(
𝑥𝑏

g𝑏−𝑥𝑏
)
𝑐

 

𝑎,𝑏,𝑐 > 0 

0 < 𝑥 < g 
[27] 

MT-II 𝐺𝑉𝐼𝐼𝐼(𝑥) = 𝑒
𝑥𝑎log2 −1  𝑎 > 0, 0 < x < 1 [28] 

Lehmann Type–I =L–I, Lehmann Type–II =L–II, Kumaraswamy=Kum, Generalized Power 

Function=GPF, Weibull Power Function=WPF, Mustapha Type–II = MT-II.  

 

6.1. Application 1 

The first data set relates to 30 measurements of tensile strength of polyester fibers discussed by 

[29]. The parameter estimates with standard errors (in parenthesis) and goodness of fit statistics are 

obtained and illustrated in Table 7. 

https://www.r-project.org/


19 Int. J. Anal. Appl. (2022), 20:23 

 

Table 7. Parameter estimates, standard errors (in parenthesis), and goodness of fit statistics for 

tensile strength of polyester fibers data. 

Model  

Parameter estimates 

(Standard errors) 
Statistics 

�̂� �̂� �̂� AIC BIC CM AD KS P-value 

ELTr-PF 
0.2169 

(0.4352) 

1.4096 

(0.7666) 
- -5.3369 -2.5345 0.0097 0.0791 0.0474 1.0000 

WPF 
3.0299 

(2.2330) 

1.3464 

(0.9412) 

0.7957 

(0.373) 
0.2444 4.4480 0.0174 0.1382 0.0611 0.9995 

Kum 
0.9627 

(0.2017) 

1.6081 

(0.4135) 
- -2.6221 0.1803 0.0183 0.1550 0.0650 0.9987 

Top-Leon 
1.1091 

(0.2024) 
- - -3.8078 -2.4066 0.0189 0.1600 0.0665 0.9981 

Beta 
0.9666 

(0.2237) 

1.6204 

(0.4106) 
- -2.6101 0.1923 0.0184 0.1559 0.0669 0.9979 

L-II 
1.6624 

(0.3035) 
- - -4.5885 -3.1873 0.0184 0.1558 0.0740 0.9924 

L-I 
0.7254 

(0.1324) 
- - -1.4495 -0.0483 0.0168 0.1425 0.1374 0.5754 

MT-II 
0.5847 

(0.1176) 
- - 0.4176 1.8188 0.0212 0.1788 0.1555 0.4201 

 
The minimum goodness of fit statistics is the criteria of a better fit model which the ELTr-PF 

distribution eventually satisfies. Hence, this research supports that the ELTr-PF distribution provides 

a better fit than its competitors. Furthermore, the curves of fitted density (a) Kaplan-Meier survival 

(b), and Probability-Probability (PP) (c) plots are presented in Figure 2. 

 

(a) 

 

(b) 

 

(c) 

Figure 2. Fitted plots for 30 measurements of tensile strength of polyester fibers data. 



20 Int. J. Anal. Appl. (2022), 20:23 

 

6.2. Application 2 

The second data set represents the failure times of 20 mechanical components studied by [30]. The 

parameter estimates with standard errors (in parenthesis) and goodness of fit statistics are obtained 

and illustrated in Table 8.  

 

Table 8. Parameter estimates, standard errors (in parenthesis), and goodness of fit statistics for the 

mechanical components data. 

Model  

Parameter estimates 

(Standard errors) 
statistics 

�̂� �̂� �̂� AIC BIC CM AD KS P-value 

ELTr-PF 
-2.9668 

(0.7335) 

2.3598 

(0.8327) 
- -74.6350 -72.6435 0.0488 0.3594 0.1054 0.9794 

Beta 
3.1119 

(0.9365) 

21.8184 

(7.0402) 
- -51.7626 -49.7711 0.3700 2.3155 0.2538 0.1520 

Kum 
1.5877 

(0.2444) 

21.8682 

(10.210) 
- -47.2969 -45.3054 0.4370 2.6508 0.2626 0.1267 

WPF 
25.3216 

(10.981) 

8.6983 

(30.616) 

0.1887 

(0.6640) 
-46.8444 -43.8572 0.3972 2.4524 0.2642 0.1226 

L-II 
7.3406 

(1.6414) 
- - -43.1863 -42.1906 0.3698 2.3142 0.3989 0.0034 

GPF 
3.1354 

(0.7011) 
- - -50.4166 -49.4209 0.4156 2.5011 0.4263 0.0014 

Top-Leon 
0.6247 

(0.1397) 
- - -25.4857 -24.4900 0.3391 2.1565 0.4842 0.0002 

L-I 
0.4484 

(0.1002) 
- - -15.1164 -14.1207 0.3211 2.0627 0.5104 0.0001 

MT-II 
0.3402 

(0.0843) 
- - -12.1937 -11.1979 0.3386 2.1538 0.5000 0.0001 

 

In Table 8, it is also clear that the ELTr-PF distribution has the lowest values for all the goodness 

of fit statistics. Therefore, the ELTr-PF distribution is recommended over its competing distributions. 



21 Int. J. Anal. Appl. (2022), 20:23 

 

The corresponding curves of fitted density (a) Kaplan-Meier survival (b), and Probability-Probability 

(PP) (c) plots are presented in Figure 3. 

 

(a) 

 

(b) 

 

(c) 

Figure 3. Fitted plots for failure times of 20 mechanical components data. 

 

7. Conclusion 

The Exponentiated Left Truncated Power (ELTr-PF) distribution has been successfully 

explored in this research. Its various statistical properties were investigated and established. The 

simulation study showed that the parameters of the ELTr-PF distribution are good and stable, as 

the root mean square error reduces as the sample size increases. The two datasets provided in this 

research support that the ELTr-PF distribution is a better fit compared to the Beta distribution, 

Kumaraswamy distribution, Lehmann Type I and Type II distributions, Generalized Power Function, 

Weibull Power Function, and Mustapha Type–II distribution. The density, Kaplan-Meier, and PP 

curves/plots also provide sufficient information about the closest fit to subject datasets. 

 

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the 

publication of this paper. 

 

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