International Journal of Analysis and Applications 

Volume 19, Number 3 (2021), 405-439 

URL: https://doi.org/10.28924/2291-8639 

DOI: 10.28924/2291-8639-19-2021-405 

 

 

Received September 1st, 2020; accepted September 16th, 2020; published April 12th, 2021. 

2010 Mathematics Subject Classification. 62E17, 05A15. 

Key words and phrases. Libby-Novick Kumaraswamy distribution; moment generating function; mean deviation; 

reliability measures; maximum likelihood estimation. 

©2021 Authors retain the copyrights 

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 

405 

 

LIBBY-NOVICK KUMARASWAMY DISTRIBUTION WITH ITS PROPERTIES AND 

APPLICATIONS 

 

ABDUL SABOOR1,*, ZAFAR IQBAL2, MUHAMMAD HANIF1, MUNIR AHMAD1 

 

1School of social sciences, National College of Business Administration & Economics, Lahore, Pakistan 

2Department of Statistics, Government College Satellite Town Gujranwala, Pakistan 

 

*Corresponding author: saboorfsd@hotmail.com 

 

ABSTRACT: The Kumaraswamy distribution is one of the most popular probability distributions with applications to 

real life data. In this paper, an extension of this distribution called the Libby-Novick Kumaraswamy (LNK) distribution 

is presented which is believed to provide greater flexibility to model scenarios involving skew-normal data than 

original one. Analytical expressions for various mathematical properties including its cdf, quantile function, moments, 

factorial moments, conditional momennts, moment generating function, characteristic function, vitality function, 

information generating function, reliability measures, mean deviations, mean residual function, Bonferroni and Lorenz 

Curves are derived.The parameters’ estimation of LNK distribution is undertaken using the method of maximum 

likelihood estimation. A simulation study for different values of sample sizes, to assess the performance of the 

parameters of LNK distribution is provided.  For illustration and performance evaluation of LNK distribution three 

real-life data sets from the field of engineering and science adapted from earlier studies are used. On comparing the 

results to previously used methods, LNK distribution shows that it can give consistently better fit than other existing 

important lifetime models. It is found that the LNK distribution is more suitable and useful to study lifetime data.  

 

 

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-19-2021-405


Int. J. Anal. Appl. 19 (3) (2021) 406 

 

 

1. INTRODUCTION 

Iqbal et al., (2017) [1] defined the Libby-Novick Kumaraswamy (LNK) distribution. The 

probability density function (pdf) and cumulative distribution function (cdf) are respectively 

defined as 

 ( )
( )

( )

1
1

 1

 1
   ;    0    1,    , ,   0

 1 1

b
a a

b
a

x x
f x abc x a b c

c x

−
−

+

−
=   

 − − 

  (1) 

( )
( )
1

  1-
1 1

b
a

a

x
F x

c x

 −
=   − − 

        (2) 

The Libby-Novick Kumaraswamy (LNK) distribution is a continuous probability distribution 

with double-bounded support. It is very similar, in many respects, to the Libby-Novick Beta 

(LNB, 1982) [2] distribution and Kumaraswamy (Kum, 1980) [3] distribution. The Kum 

distribution is a two parameter distribution like Beta distribution Where as M.C. Jones (2009) [4] 

find out numerous benefits of the Kum distribution over classical beta distribution.The Kum 

distribution is a special case of McDonald’s (1984) [5] generalized Beta of the first kind 

distribution. One key difference between the Kum and Beta distributions is the availability for 

the former, but not for the latter, of an invertible closed form cumulative distribution function 

presented by Mitnick (2013) [6]. 

The two distributions LNK and LNB are very flexible and can take approximately the 

same shapes and the former distribution is also a generalized distribution of Kum distribution. 

There are, however, important realistic differences between LNB and LNK distributions. On the 

one hand, the availability for the LNK , but not for the LNB distribution, of an invertible closed-

form cumulative distribution function makes the LNK distribution much better suited than the 

LNB for activities that require the generation of random varieties, in particular simulation 

modeling and simulation-based model estimation. In contrast, the lack of tractable-enough 

expressions for the mean and variance of the LNK distribution has stalled its utilization for 

modeling purposes; in spite of the advantages that the availability of an invertible closed-form 

cumulative distribution function entails, the LNK distribution may use rather sparingly in the 

modeling of stochastic phenomena and processes. Libby-Novick (1982) [2] derived two new 

multivariate probability density functions, which are the generalized forms of beta distribution 



Int. J. Anal. Appl. 19 (3) (2021) 407 

 

 

and F distribution. They proved that these distributions that seem to be suitable in utility 

modeling. They reproduced the same distributional form in the cases of marginal and conditional 

distributions. Chen and Novick (1984) [7] used the Libby-Novick (1982) [2] univariate generalized 

beta distribution as a prior and estimated the parameters of Bernoulli and Binomial distributions. 

These expressions are the generalized forms of the standard beta class. Ristic et al., (2013) [8] 

derived new family of skewed distributions such as Libby and Novick’s generalized beta 

exponential distribution and found some useful properties of this family of distributions. 

Cordeiro et al., (2014) [9] defined a family of distributions, named the Libby-Novick beta family 

of distributions, which includes the classical beta generalized and exponentiated generators. This 

extended family gave reasonable parametric fits to real data in several areas because the 

additional shape parameters controlled the skewness and kurtosis simultaneously. Ali (2019) [10] 

worked on new form of Libby-Novick (NLN) distribution and explored some properties of NLN 

distribution. This model was compared with other distributions by fitting them to a real data set.  

Rashid et al., (2020) [11] derived different entropy measures and characterized Libby-Novick 

generalized Beta (LNGB) distribution through various methods. Iqbal et al., (2021) [12] derived 

mathematical properties of LNGB distribution and applied it to modeling on three real data sets.  

In this paper we derive basic and advanced properties of LNK distribution and find applications 

of LNK distribution to three real data sets. In Section 2, a detailed remarks about the graphs of 

pdf, hazard rate function, reverse hazarz rate function and survival function are provided.  

Section 2 also contains the derivation and results of some important mathematical properties of 

LNK distribution. Maximum likelihood estimators of three parameters of LNK distribution are 

derived in section 3. In section 4, a simulation of the parameters is carried out for different sample 

sizes. A number of deduced models are shown in section 5. In section 6, LNK distribution is 

compared with some other models through three data sets. Some concluding remarks are 

presented in section 7. 

 

2. MATHEMATICAL PROPERTIES 

1.1 Shape properties of the pdf 

The LNK density function defined in (1) has real flexibility and it is shown through graphs 

w.r.t. some different combinations of values of the parameters. 



Int. J. Anal. Appl. 19 (3) (2021) 408 

 

 

 

      

 

 

 

 

 

 

 

 

 

         

    

 

 

 

 

 

 

 

 

 

         

a 0.5 ,b 0.5 ,c 1

a 5.0 ,b 1.0 ,c 1

a 1.0 ,b 3.0 ,c 1

a 2.0 ,b 5.0 ,c 1

a 2.0 ,b 2.0 ,c 1

0.2 0.4 0.6 0.8 1.0
x

1

2

3

4

f x

a 0.5 ,b 0.5 ,c 1.5

a 5.0 ,b 1.0 ,c 1.5

a 1.0 ,b 3.0 ,c 1.5

a 2.0 ,b 5.0 ,c 1.5

a 2.0 ,b 2.0 ,c 1.5

0.2 0.4 0.6 0.8 1.0
x

1

2

3

4

f x

a 0.5 ,b 0.5 ,c 0.5

a 5.0 ,b 1.0 ,c 0.5

a 1.0 ,b 3.0 ,c 0.5

a 2.0 ,b 5.0 ,c 0.5

a 2.0 ,b 2.0 ,c 0.5

0.2 0.4 0.6 0.8 1.0
x

0.5

1.0

1.5

2.0

2.5

3.0

3.5

f x



Int. J. Anal. Appl. 19 (3) (2021) 409 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.1 pdf Graph of LNK distribution 

Shapes of pdf 

(i) For a > 0, c > 0 and 0 < b < 1 then LNK distribution is U-shaped. 

(ii) For 0 < a < 1, b > 1 and 0 < c < 1 then LNK distribution is S-shaped. 

(iii) For a = b = 2, c ≥ 1 (c < 1) and if c →  (c → 0) then LNK distribution increases its positively 

skewed (negatively skewed) from the symmetric with decreasing mode (increasing 

mode). 

(iv) For a = b = c = 1, the LNK distribution is uniform distribution. 

(v) For a > 2, b > 2 and c ≥ 1 the LNK distribution is unimodel positively skewed with 

decreasing mode when c→ . 

(vi) For a > 2, b = 1, 0 < c < 1, the form of LNK distribution is an increasing. 

a 0.5 ,b 0.5 ,c 0.1

a 5.0 ,b 1.0 ,c 0.1

a 1.0 ,b 3.0 ,c 0.1

a 2.0 ,b 5.0 ,c 0.1

a 2.0 ,b 2.0 ,c 0.1

0.2 0.4 0.6 0.8 1.0
x

1

2

3

4

f x

a 0.5 ,b 0.5 ,c 0.5

a 5.0 ,b 1.0 ,c 0.5

a 1.0 ,b 3.0 ,c 2.0

a 2.0 ,b 2.0 ,c 3.0

a 2.0 ,b 5.0 ,c 0.1

0.2 0.4 0.6 0.8 1.0
x

1

2

3

4

f x



Int. J. Anal. Appl. 19 (3) (2021) 410 

 

 

(vii) For a > 2, b = 1, 1< c < 3.5, the LNK distribution increase but it increases slowly when c 

increases in the interval and for c ≥ 3.5 the LNK distribution again turns to unimodel.  

(viii) For b = c = 1, the LNK distribution is Power distribution. 

(ix) For a = c = 1, the LNK distribution is LNK is a special case of Kum-distribution or reflected 

exponentiated distribution. 

1.2 Distribution function 

The cumulative distribution function (cdf) of LNK distribution is  

( )
( )

( )

1
1

 1

0

 1
   

 1 1

b
a ax

b
a

t t
F x abc dt

c t

−
−

+

−
=

 − − 
      (3) 

By making substitutions 
1  + cz

a c z
t z and u

z
= =

−                           (4)
 

And after some simplification we have the expression of cdf as shown in (2) 

 

 ,  

 

 

 ,  

Figure 2.2 cdf Graph of LNK distribution 

0.2 0.4 0.6 0.8 1.0
x

0.2

0.4

0.6

0.8

1.0

F x

a 0.5,b 0.5,c 0.5

a 5.0,b 1.0,c 0.5

a 1.0,b 3.0,c 2.0

a 2.0,b 2.0,c 3.0

a 2.0,b 5.0,c 0.1

0.2 0.4 0.6 0.8 1.0
x

0.2

0.4

0.6

0.8

1.0

F x

a 0.5,b 0.5,c 3.0

a 0.5,b 1.5,c 0.1

a 2.0,b 2.0,c 0.5

a 2.0,b 5.0,c 2.0

a 3.0,b 2.0,c 0.3



Int. J. Anal. Appl. 19 (3) (2021) 411 

 

 

2.3   Quantile Function 

The quantile function of LNK distribution is given by: 

( )F x p=   Where  0 < p < 1 

And it 
1
( )x F p

−
=   

1
1

1 (1 )

b
a

a

x
p

c x

 −
− = 

− − 
 

After simplification we find the quantile function of the LNK distribution as under; 

( )
1/

1/

1/

1 (1 )
;  ,  , (0,1)

1 (1 )(1 )
 

a
b

b

p
Q p a b c p

c p

 − −
=  

− − − 
 

Where ( );  ,  ,Q p a b c  is the inverse of LNK function or quantile of LNK function at p. Clearly, the 

function has explicit form and it can be numerically solved through software for different set of 

parameters’ values.The graph can also be used to illustrate the behaviour of quantile function of 

LNK distribution. 

i. For a > 1, b > 1 by increasing c, the quantile value of the LNK distribution decreses 

comparatively. 

ii. For 1,a 
 
c > 1, and for any value of b, the quantile value of the LNK distribution increases 

slowly for 0.5p   and for 0.5p  , the quantile value sharply increases. 

iii. For c → 0, the quantile value of the LNK distribution increases sharply. 

2.4      Hazard Rate Function 

The hazard or instantaneous rate function is denoted by h(x). The hazard function of x can be 

interpreted as instantaneous rate or the conditional probability density of failure at time x, given 

that the unit has survived until x. The hazard rate function of LNK distribution as  

( )

1

( ; , , )
1 (1 ) 1

a

a a

abcx
h x a b c

c x x

−

=
 − − − 

 

 



Int. J. Anal. Appl. 19 (3) (2021) 412 

 

 

      

 

 

 

Figure 2.3 hazard rate function Graph of LNK distribution 

a 0.5,b 0.5,c 0.5

a 5.0,b 1.0,c 0.5

a 1.0,b 3.0,c 2.0

a 2.0,b 2.0,c 3.0

a 2.0,b 5.0,c 0.1

0.2 0.4 0.6 0.8 1.0
x

5

10

15

20

25

h x

a 0.5,b 0.5,c 3.0

a 0.5,b 1.5,c 0.1

a 2.0,b 2.0,c 0.5

a 2.0,b 5.0,c 2.0

a 3.0,b 2.0,c 0.3

0.2 0.4 0.6 0.8 1.0
x

5

10

15

20

25

h x

a 0.5,b 0.5,c 0.5

a 5.0,b 1.0,c 0.5

a 1.0,b 3.0,c 2.0

a 2.0,b 2.0,c 3.0

a 2.0,b 5.0,c 0.1

0.2 0.4 0.6 0.8 1.0
x

5

10

15

20

h x



Int. J. Anal. Appl. 19 (3) (2021) 413 

 

 

The hazard rate function of LNK distribution is of Bath-Tub shape, increasing shape and 

decreasing shape for specific sets of values of parameters.  

2.5     Reverse Hazard rate function 

The reverse hazard can be interpreted as an approximate probability of failure in

 ,   + dx x , given that the failure has occurred in 0,  x . The reverse hazard function 

( ; , , )r x a b c  of LNK distribution is defined as 

            

( ) ( ) ( )( )
1 1
(1 )

( ; , , )
1 (1 ) 1 (1 ) 1

a a b

b b
a a a

abcx x
r x a b c

c x c x x

− −
−

=
− − − − − −

  

   

 

Figure 2.4 reverse hazard rate Graph of LNK distribution 

a 0.5,b 0.5,c 0.5

a 5.0,b 1.0,c 0.5

a 1.0,b 3.0,c 2.0

a 2.0,b 2.0,c 3.0

a 2.0,b 5.0,c 0.1

0.2 0.4 0.6 0.8 1.0
x

5

10

15

20

25

30

r x

a 0.5,b 0.5,c 3.0

a 0.5,b 1.5,c 0.1

a 2.0,b 2.0,c 0.5

a 2.0,b 5.0,c 2.0

a 3.0,b 2.0,c 0.3

0.2 0.4 0.6 0.8 1.0
x

5

10

15

20

r x



Int. J. Anal. Appl. 19 (3) (2021) 414 

 

 

2.6  Survival Function  

    The survival function or reliability function of LNK distribution is defined as  

           ( )
1

1 (1 )

b
a

a

x
S x

c x

 −
=  

− − 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.5 survival function Graph of LNK distribution 

 

 

a 0.5,b 0.5,c 3.0

a 0.5,b 1.5,c 0.1

a 2.0,b 2.0,c 0.5

a 2.0,b 5.0,c 2.0

a 3.0,b 2.0,c 0.3

0.2 0.4 0.6 0.8 1.0
x

0.2

0.4

0.6

0.8

1.0

S x

a 0.5,b 0.5,c 0.5

a 5.0,b 1.0,c 0.5

a 1.0,b 3.0,c 2.0

a 2.0,b 2.0,c 3.0

a 2.0,b 5.0,c 0.1

0.2 0.4 0.6 0.8 1.0
x

0.2

0.4

0.6

0.8

1.0

S x



Int. J. Anal. Appl. 19 (3) (2021) 415 

 

 

2.7  The rth Moment 

The rth moments about origion of the LNK distribution is defined as 

1

0

( ) ( )
r r

E x x f x dx=    

By Substitution (4) becomes 

 

1 1

1

0

(1 )

1 (1 )

r

ba

b

bcy y
dy

c y

−

+

−
=

− −
  

1

1

0

(1 )

r

a
bu

b u du
c u uc

− 
= − 

+ − 
   

( )
( )

1

1

00

/ 1
(1 ) 1

/ !

kr r

b ka a

k

r a k
bc u u u du

r a k c

− 
−

=

 +  
= − − 

  
  

( )
( )

1

1

0 0

/ 1
1 (1 )

/ !

kr r
k

ba a

k

r a k
bc u u du

r a k c

− 
+

−

=

 +  
= − − 

  
   

( )
( )

( )
/

0

/ 1
( ) 1 / 1,

/ !

k

r

r a
k

r a kb
E x B r a k b

c r a k c



=

 +  
= − + + 

  
  

Corollary 

( )
( )

( )
1/

0

1 / 1
( ) 1 1 / 1,

1 / !

k

a
k

a kb
E x B a k b

c a k c



=

 +  
= − + + 

  
  

( )
( )

( )2
2/

0

2 / 1
( ) 1 2 / 1,

2 / !

k

a
k

a kb
E x B a k b

c a k c



=

 +  
= − + + 

  
  

 

2.8    Moment generating function 

The moment generating function of the LNK distribution about zero is 

( ) ( )tM t  = E e xx
        

( )
 = 0 0

t1
= 1 1,

! !

r
rk

a

r k

bc r r
k B k b

r a c a k r

a

−

 

=

    
 + − + +    

      
 
 

    
0

e
!

r
x

r

x

r



=

=   



Int. J. Anal. Appl. 19 (3) (2021) 416 

 

 

2.9   Factorial Moments  

This section devotes to inceasing and decreasing factorial moments of LNK distribution as 

under:  

2.9.1      Decreasing Factorial Moments of LNK distribution  

The decreasing factorial moments of the LNK distribution is defined as:  

( ) ( )
n

r

r = 0

= s n, r E X            
( )n

r = 0 0

n, r 1
= 1 1,

!

kr

a

k

s r r
bc k B k b

r a c a
k

a

−

=

    
 + − + +    

      
 
 

  

Where ( )s n, r  are the Stirling’s numbers of first kind.
 

2.9.2 Increasing Factorial Moments of LNBD  

The increasing factorial moments of the LNK distribution is defined as: 

 

 

    ( ) ( )
n

r

r = 0

= n, r E Xd  

               
( )n

r = 0 0

n, r 1
= 1 1,

!

kr

a

k

d r r
bc k B k b

r a c a
k

a

−

=

    
 + − + +    

      
 
 


   

 

Where  ( )n, rd  can be deduced from the relation 

   

2.10    Negative Moments 

The negative moments of the LNK distribution is defined as:  

( ) ( )
1

r r

0

E X  = ;  a, b dx f x x
− −

   

By applying the substitution (4), we have 

( )r
0

1
E X  = ( 1) 1 1,

kr

ka

k

n r
bc B k b

k c a


−

=

     
− − − + +     

    
  

 



Int. J. Anal. Appl. 19 (3) (2021) 417 

 

 

2.11 Incomplete Moments 

 The Incomplete moments of the LNK distribution is defined as: 

0

( ) ( )

x

r

r
x x f x dx =    

1 1

1

0

(1 )

1 (1 )

x r a a b

b
a

x abcx x
dx

c x

− −

+

−
=

 − − 
   

0

1
1 ( ; 1, )

1
!

k a

r a a
ka

r
k

b cx ra
I k b

r c x cx a
kc

a



=

 
 + 

  
= − + + 

− +   
 
 

  
 

2.12   Scaled Total Time for Aging Properties  

The scaled total time of the LNK distribution is defined as: 

( )( ) ( ) ( )
1

0 0

1 1
x x

F

y

S F x S y dy f t dt dy
 

= =   

( )( ) ( )
0

1
x

F
S F x x F y dy



 
= − 

 
  

( )( )
0

1 1

1 (1 )

bx a

F a

y
S F x dy

c y

 −
=  

− − 
  

2.13  Conditional Moments 

The conditional moments of the LNK distribution is defined as: 

 

( )
1

r

r

1
M  = t t dt

F
x

f     

By applying the substitution (4) and after some simplification, we have

 

( )( )

( )
r

0

1 1 1
M  = 1 ( ; 1, )

11 !

b
a k a

rb r a a
a

ka

r
k

c x b cx ra
I k b

r c x cx ax kc
a




=

  
 +  − −     − − + + 

− +    −     


  

2.14 Mean Residual Function 

The mean residual function of the LNK distribution is defined as: 

r

r
M  = E X   X > x 

 



Int. J. Anal. Appl. 19 (3) (2021) 418 

 

 

( )m  = E X   X > x x x −   

( )m  = E X  X >   x x x  −   

          
( )( )

( )
1 1

0

1
1 1 1 1

= 1 ( ; 1, )
1 11 !

b
a k a

b a a
a

ka

k
c x b cxa

I k b x
c x cx ax kc

a




=

  
 +  − −    − − + + − 

− +    −     


  

2.15 Vitality Function 

The vitality function of the LNK distribution is defined as: 

( )V  = E X X > x x    

( )
( )( )

( )
1 1

0

1
1 1 1 1

V  = = 1 ( ; 1, )
1 11 !

b
a k a

b a a
a

ka

k
c x b cxa

x I k b
c x cx ax kc

a




=

  
 +  − −    − − + + 

− +    −     

  

2.16 Geometric Vitality Function 

The Geometric vitality function of the LNK distribution is defined as: 

( )logG  = E log X X > x x    

( )
( )

( )
1

1
logG  = log t  t dt

1 F x
x

x f
−

  

( )
( ) ( )

1

0 0

1
 = log t  t dt- log t  t dt

1 F x

x

f f
 
 

−  
   

( )
( ) ( )( ) ( ) ( )

  1 0

1 1 ln c 1 1
1 1 +  1  B 1, log t  t dt

1 F x c

i x

i

b b i b f
a a i
 



=

   
= − − + − + −    −     

   

( )

( ) ( )( ) ( )
( )

( )

1 1

1

  1 0

1 1

1

0

1 ln c 1 1
1 1 +  1  B 1, (1 ) (ln )

c1

1 F x

(1 ) ln( )

a

a

a

a

cx

c xi

b

i

cx

c x

b

b
b b i b u u du

a a i a

b
u c u uc du

a

 

− −


−

=

− −

−

 
   

− − + − + − −   
    

=  
−

 
 
+ − + − 
 

 



 

 

 



Int. J. Anal. Appl. 19 (3) (2021) 419 

 

 

2.17 Characteristics Function 

The characteristic function of the LNK distribution is defined as:  

( ) ( )tt  = E ei xx    

                      

( )

  0 0

1
 1 1,

! !

r
rk

a

r k

itbc r r
k B k b

r a c a k r

a

−

 

= =

    
=  + − + +    

      
 
 

    
0

e
!

r
x

r

x

r



=

=   

2.18 Information generating function 

The information generating function of the LNK distribution is defined by P(s) and found 

from 

( ) ( )1sP s E f −=  

        
( )

( )

1
11

 b+1

0

 1
abc  dx

 1 1

s
b

a a

a

x x

c x

−
− −

 =
  − −  

  

By applying the substitution (4), we have 

    ( )
( ) 11 ( 1)/ ( 1)(s 1)/ ( 1)

0 0

( 1)(1 ) / 1
1 (1 )

(( 1)(1 ) / ) !

i

s s s a a a s b i

i

a s a i
P s a b c u u u du

a s a i c


− − − − −

=

 + − +  
= − − 

 + −  
   

( ) 11 ( 1)/ ( 1)( 1)/ ( 1)
0 0

( 1)(1 ) / 1
1 (1 )

(( 1)(1 ) / ) !

i

s s s a a s a i s b

i

a s a i
a b c u u du

a s a i c


− − − − + −

=

 + − +  
= − − 

 + −  
   

      
( )

( )1 ( 1)/
0

( 1)(1 ) / 1
1 ( 1)( 1) / 1, ( 1) 1

(( 1)(1 ) / ) !

i

s s s a

i

a s a i
a b c B a s a i s b

a s a i c


− −

=

 + − +  
= − − − + + − + 

 + −  
  

2.19            Some Other Measures of Averages 

2.19.1    Harmonic Mean 

This section contains Harmonic mean of x for the LNK distribution 

The Harmonic mean of x is as under: 

1

1
x

H

E
x

=
 
 
 

  

Consider 

1

0

1 1
( )E f x dx

x x

 
= 

 
   



Int. J. Anal. Appl. 19 (3) (2021) 420 

 

 

1 2 1

1

0

(1 )

[1 (1 ) ]

a a b

a b

x x
abc dx

c x

− −

+

−
=

− −
   

By substituting
a

x y= , we have 

1
1 1

1

0

(1 )

[1 (1 ) ]

ba

b

bcy y
dy

c y

−
−

+

−
=

− −
   

After substitution (4), and simplification, we have 

1
1 1

1

0

(1 )

[ ]

ba

a

bu u
du

c u uc

−

−

−

−
=

+ −
   

1 1 1

1

0

(1 ) ( )
ba abu u c u uc du

−

−
= − + −   

11 1

1

00

1 / 1
(1 ) (1 )

i

ba a

i

a
bc u u u du

i c

− 
−

=

  
= − − −  

  
   

1

0

1 /1 1 1
(1 ) ( 1, )

i

a

i

a
E bc B i b

ix c a



=

  −   
= − − + +    

    
  

1

0

1
1.

1 / 1 1
(1 ) ( 1, )

ix

a

i

H for a
a

bc B i b
c ai



=

= 
  − 

− − + +  
  



  

Corollary 

If c=1, Then 
1

1
1,

x
H

bB b
a

=
− 

+ 
 

 is the Harmonic Mean of Kum distribution 

2.19.2     Geometric Mean 

The logarithm of the geometric 
X

G of a distribution with random variable X is the arithmetic 

mean of( )ln X , or, equivalently, its expected value ( )ln lnXG E X= . 

1

0

(ln ) (ln ) ( )E x x f x dx=    



Int. J. Anal. Appl. 19 (3) (2021) 421 

 

 

1 1 1

1

0

(1 )
(ln ) (ln )

1 (1 )

a a b

b
a

abcx x
E x x dx

c x

− −

+

−
=

 − − 
  

By substitution 
a

x y= and (4), after simplification, we have  

( ) ( )( ) ( )
  1

1 ln c 1 1
 = exp 1 1 +  1  B 1, 

c

i

X

i

G b b i b
a a i
 



=

  
− − + − +  

   
  

2.19.3 Mode 

Mode is obtained by solving ( )
d

 = 0
d

f x
x

 

( )
( )

( ) ( )

b 1a a 1

a + b

c  1
 = 

B a, b  1 1 c

−−
−

− −  

x x
f x

x
 

Mode of LNK distribution is obtained by solving 

( ) 0
d

f x
dx

=   

1 1
1 ( 1) (1 )( 1)

'( ) { } ( )
(1 ) [1 (1 ) ]

a a

a a

a a b x a b c x
f x f x

x x c x

− −
− − + −

= − −
− − −

 

1 1
1 ( 1) (1 )( 1)

0
(1 ) [1 (1 ) ]

a a

a a

a a b x a b c x

x x c x

− −
− − + −

− − =
− − −

 

Multiplying both sides by x 

( 1) (1 )( 1)
1 0

(1 ) [1 (1 ) ]

a a

a a

a b x a b c x
a

x c x

− + −
− − − =

− − −
 

Put 
a

x y=   

( 1) y (1 )( 1)
1 0

1 1 (1 )

a b a b c
a

y c y

− + −
− − − =

− − −
  

( 1)(1 y)(1 y cy) {a(b 1) y(1 y cy)} {a(1 b)(c 1) y(1 y)} 0a − − − + − − − + − + − − =  

( )( ) ( )22 (1 ) 2 1 1 0a c y c ab y a− − + − + + − =  
2

0Ay By C+ + =  

  
2

 B  B   4AC
y = 

2A

−  −
 



Int. J. Anal. Appl. 19 (3) (2021) 422 

 

 

1/
2

 B  B   4AC
x = 

2A

a

 −  −
 
 
 

 

Where 
a

y x=   

2 (1 )A a c= − −   

2B c abc= − −   

( 1)C a= −   

2.20  Points of Inflection 

The points of the inflection of the LNK distribution are found from  

''
( ) 0f x =  under 

'''
( ) 0f x   or equavalentally as 

2

2

ln ( )
0

d f x

dx
=  and 

3

3

ln ( )
0

d f x

dx
  

Where as  

2 2

2 2 2

2

2

ln ( ) ( 1) ( 1) [( 1) ]

(1 x )

(1 )(1 ) [( 1) (1 ) ]

[1 (1 ) ]

a a

a

a a

a

d f x a a b x a x

dx x

a b c x a c x

c x

−

−

− − − − +
= − +

−

+ − − + −

− −

  

So  

2

2 2

( 1) ( 1) [( 1) ]

(1 x )

a a

a

a a b x a x

x

−
− − − − +

−
−

+

2

2

(1 )(1 ) [( 1) (1 ) ]

[1 (1 ) ]

a a

a

a b c x a c x

c x

−
+ − − + −

− −
=0 

After some Simplification 

4 3 2
0Ay By Cy Dy E+ + + + =   

Where 
a

y x=   

2
( 1)(1 )A a c= − + −   

2
(1 )[(2 )( 2 ) (1 )]B c c a a abc a= − − + − − +   

2 2
(1 )(4 6) ( 2)C c a abc c c= − − − − −   

( 1)[( 2)( 2) ]D a c a abc= − − + +   

( 1)E a= −   



Int. J. Anal. Appl. 19 (3) (2021) 423 

 

 

Table 2.1 Points of inflection of LNK distribution 

A B C Point (s) of inflection 

1 2 0.1 0.8482 

1 2 0.2 0.6585 

1 2 0.3 0.4146 

1 2 0.4 0.0894 

1 3 0.1 0.7318 

1 3 0.2 0.3965 

2 3 0.1 0.4432 , 0.8678 

2 3 0.2 0.5681 , 0.6607 

2 4 0.1 0.4625, 0.8089 

2 5 0.1 0.4888, 0.7454 

2 6 0.1 0.5342, 0.6644 

3 4 0.1 0.5492, 0.8741 

3 5 0.1 0.5665, 0.8335 

3 6 0.1 0.5897, 0.7881 

3 7 0.1 0.6294, 0.7273 

 

2.21 Bonferroni Curve 

The Bonferroni curve of the LNK distribution is as:   

( )( )
( )

( )
0

1
x

F
B F x y f y dy

F x
=   

( )( )

( )

0

0

1

1 1
1 ( ; 1, )

1 1
!

 = 
1

1 1 1
1- 1 1,

11 1
!

k a

a a
k

F

b ka

a
k

k
cxa

I k b
c x cx a

k
a

B F x

k
x a

B k b
c x c a

k
a



=



=

 
 + 

  
− + + 

− +   
 
 

 
 +   −       − + +     

 − −           
 





 

 

2.22 Lorenz Curve 

The Lorenz curve of the LNK distribution is as: 

( )( ) ( ) ( )( )L F x F x B F x=  



Int. J. Anal. Appl. 19 (3) (2021) 424 

 

 

( )( )

0

0

1

1 1
1 ( ; 1, )

1 1
!

1

1 1
1 1,

1
!

k a

a a
k

k

k

k
cxa

I k b
c x cx a

k
a

L F x

k
a

B k b
c a

k
a



=



=

 
 + 

  
− + + 

− +   
 
 

=
 

 + 
    
− + +   

     
 
 



  

2.23 Gini Coefficient 

The Gini coefficient of the LNK distribution is explained as: 

( )( )
1

21

0

1  1G F x dx
−

= − −  

    
( )

2
1

0

1 1
1  

1 1

b
a

a

x
dx

c x

 −
= −   − − 

  

2.24 Mean deviations 

The mean deviation of the LNK distribution from arithmetic mean and median are 

denoted by ( )1δ Y  and ( )2δ Y  respectively and are found from 

( ) ( )
1

1

0

δ  =   Y y f y dy−  

( ) ( ) ( ) ( )
1 1

0 0

 =      +     f y dy y f y dy y f y dy f y dy

 

 

 − −     

( ) ( ) = 2    2 F J  −  

( )
1

0

1

1 1 1
 = 2  1   2 1 ( ; 1, )

11 1 1
!

b ka a

a a a
ka

k
b ca

I k b
c c c a

kc
a

 


  



=

 
 +   −    − − − + +     − − − +        
 

  

The mean deviation about median 

         ( ) ( )
1

2

0

δ  =   Y y M f y dy−  

        ( ) ( ) ( ) ( )
1 1

0 0

 =      +     

M M

M M

M f y dy y f y dy y f y dy M f y dy− −     



Int. J. Anal. Appl. 19 (3) (2021) 425 

 

 

        ( ) =   2 J M −  

        
1

0

1

1 1
 =   2 1 ( ; 1, )

1 1
!

k a

a a
ka

k
b cMa

I k b
c M cM a

kc
a




=

 
 + 

  
− − + + 

− +   
 
 

  

 

3. ESTIMATION 

Here, we consider 

 

3.1 Maximum-likelihood estimation 

 

The likelihood function of random sample 1 2, ,..., nx x x  of observation is given by 

1 2

1

( , ,..., ;a, b,c) ( , , , )
n

n i

i

f x x x f x a b c
=

=   

1

( , , , ) ( , , , )
n

i i

i

L x a b c f x a b c
=

=     

( )

( )

1 1

1 1

1

1

abc (1 )

( , , , )

1 (1 )

a b
n n

n a

i i

i i

i b
n

a

i

i

x x

L x a b c

c x

− −

= =

+

=

   
−   

   
=

 
− − 

 

 



  

Taking ln both sides 

( )
1 1 1

ln ln ( 1) ln (b 1) ln(1 ) (1 ) ln 1 (1 c)
n n n

a a

i i i

i i i

L n abc a x x b x
= = =

= + − + − − − + − −     

It follows that the maximum-likelihood estimates ( )a, b, c , say ( )ˆ垐a, b, c , are the simultaneous 

solutions of the equations: 

( )1 1 1
ln (1 ) ln

ln ( 1) (1 )
(1 ) 1 (1 c) x

a an n n
i i i i

i a a
i i ii i

L n x x c x x
x b b

a a x= = =

 −
= + − − + +

 − − −
     

( )
1 1

ln(1 ) ln 1 (1 c) x
n n

a a

i i

i i

L n
x

b b = =


= + − − − −


    



Int. J. Anal. Appl. 19 (3) (2021) 426 

 

 

1

(1 )
1 (1 c) x

an
i

a
i i

L n x
b

c c =


= − +

 − −
   

For the ML estimators, we solve the following three equations simultaneously we have 

0, 0, 0
L L L

a b c

  
= = =

  
 

Since the equations are of implicit and have the complicated forms. Therefore, for numerically 

solution of the equations we can use differenent softwares like The R Mathematica 10.2 etc. 

 

4. SIMULATION STUDY 

Table 1 The Bias, MSE values for the LNK model when a=b=c=2.0 

 n=10 n=20 n=30 n=50 n=70 n=100 

Bias(a) 0.6789 0.2966 0.1591 0.0732 0.0398 0.0134 

Bias(b) 148.4399 98.1754 80.9118 56.4929 42.0678 37.0448 

Bias(c) 519.4312 63.2427 5.1913 0.3873 -0.1171 -0.6091 

MSE(a) 2.3026 0.7537 0.3908 0.1723 0.1132 0.0843 

MSE(b) 45131.8214 22559.3813 17215.8524 9705.1161 5973.0134 5221.2081 

MSE(c) 4729852.1234 592121.2123 17854.2713 33.3845 15.21264 4.6655 

 n=150 n=200 n=300 n=500 n=700 n=1000 

Bias(a) 0.0165 0.0013 0.0025 -0.0071 -0.0157 -0.0134 

Bias(b) 25.9659 17.9847 12.0603 6.9705 5.6428 4.7767 

Bias(c) -0.7730 -0.9022 -0.9606 -0.9643 -0.9604 -0.9707 

MSE(a) 0.0578 0.0445 0.0286 0.0179 0.0123 0.0081 

MSE(b) 2688.8471 1429.0832 685.7248 176.8713 80.2722 36.4003 

MSE(c) 2.2229 1.8662 1.5139 1.3256 1.2007 1.1071 

 

 

 

 

 

 

 

 



Int. J. Anal. Appl. 19 (3) (2021) 427 

 

 

Table 2 The Bias, MSE values for the LNK model when a=b=c=2.5 

 n=10 n=20 n=30 n=50 n=70 n=100 

Bias(a) 0.7101 0.1423 0.0226 -0.0336 -0.0764 -0.0892 

Bias(b) 148.6944 83.5053 70.3768 49.9135 40.9383 28.8218 

Bias(c) 908.6814 240.0717 101.7437 6.0567 3.3570 1.6298 

MSE(a) 2.5889 0.6457 0.4488 0.2476 0.1627 0.1103 

MSE(b) 46025.2324 19184.2612 13459.1232 7983.5764 5993.6893 3566.5871 

MSE(c) 8932954 2172753 858508 1049.3921 181.7757 45.5042 

 n=150 n=200 n=300 n=500 n=700 n=1000 

Bias(a) -0.1165 -0.1040 -0.1170 -0.1267 -0.1286 -0.1309 

Bias(b) 16.4083 11.5937 8.2968 4.7545 3.6887 3.1612 

Bias(c) 0.9505 1.0242 0.6185 0.4136 0.4096 0.2923 

MSE(a) 0.0804 0.0619 0.0491 0.0373 0.0310 0.0282 

MSE(b) 1369.2381 791.9238 402.1093 108.3226 41.8772 16.9204 

MSE(c) 15.8173 13.1791 6.4383 3.0282 2.2463 1.4012 

Table 3 The Bias, MSE values for the LNK model when a=b=c=3.0 

 n=10 n=20 n=30 n=50 n=70 n=100 

Bias(a) 0.4603 0.0535 -0.0688 -0.1884 -0.2438 -0.2725 

Bias(b) 139.5197 81.7559 59.6459 39.1355 28.2413 16.5283 

Bias(c) 1318.3864 540.5148 167.8884 61.5224 12.9443 9.0277 

MSE(a) 2.1234 0.9155 0.6307 0.3236 0.2394 0.2077 

MSE(b) 43208.6434 18732.8834 11124.5823 5996.8751 3686.9514 2041.9723 

MSE(c) 12800727 4713924 1058158 285476.4 1760.2651 455.9306 

 n=150 n=200 n=300 n=500 n=700 n=1000 

Bias(a) -0.2895 -0.3107 -0.3037 -0.3063 -0.3198 -0.3133 

Bias(b) 10.2456 6.4328 3.8052 2.2502 1.5573 1.3494 

Bias(c) 7.5233 7.4066 5.9516 5.0419 5.2533 4.9514 

MSE(a) 0.1698 0.1596 0.1384 0.1194 0.1204 0.1101 

MSE(b) 845.9625 368.9467 139.7221 35.0732 8.4393 3.1515 

MSE(c) 238.0884 154.0213 83.8912 46.6563 42.8393 34.4719 



Int. J. Anal. Appl. 19 (3) (2021) 428 

 

 

5. SOME TRANSFORMED DISTRIBUTIONS 

 

5.1 Following table show the dffierent transformation of LNK distribution: 

Sr. No. a b c Transformation Resulting Distribution 

1 a b 1 X Kumaraswamy distribution (1980) 

2 a b 1 1 – x 

Corderio (2012) distribution  

( ) ( )( ) = 1- 1 1 ;   
0 <  < 1,   a,b > 0

b
a

F x x

x

− −

 

3 1 b 1 X 
B(1,b) and Kum(1,b) 

Power distribution 

4 a 1 1 X 
B(a,1) and Kum (a,1) 

Power distribution 

5 a 1 1 -logx Exp(a)  

6 1 b 1 -log(1-x) Exp(b) 

7 a b 1 X Kum(a,b) or G 1B  (a,1,1,b) 

8 1 b 1 
1

x

x+
 

( )
( )

1
 1

1
b

F x
x

= −
+

0 <  < ,   b > 0x 

 

9 1 b c 
1

cx

x cx− +
 

( ) ( ) = 1 1 ;   

0 <  < 1,   b > 0

b
F x x

x

− −

 

10 1 1 1 X Uniform (0,1) 

 

5.2 Asymptotes and Shapes 

The asymptotes of (1.1), (1.2) and (3.1) as 0,1x →  are given by 

 ( )  pdf f x  ( )  hrf h x  

0x →  
a 1

, , 1a b c x where a b c
−

  
a 1

, , 1a b c x where a b c
−

  

1x →  ( )
b 1

1 , , 1
a

a b c x where a b c
−

−   
( )1 a

abc

x−
, , 1where a b c   

 



Int. J. Anal. Appl. 19 (3) (2021) 429 

 

 

6. APPLICATION 

In order to prove that LNK distribution can be a better model than the Power distribution, 

Beta distribution with (a = 1), Beta distribution, Kumaraswamy distribution, let us use three real 

data sets. 

The following tables show the numerical values with MLEs and their corresponding 

standard errors (in parentheses) of the model parameters including loglikelihood, Kolmogorov-

Smirnov test (KS), Akaike information criterion (AIC) and Consistent Akaike information 

criterion (CAIC) for comparing LNK distribution with the Power distribution, Beta distribution 

with (a =1), Beta distribution, Kumaraswamy distribution. It is quite evident from the reports that 

LNK distribution is better. The plots of the fitted distributions to real datasets are shown in 

figures. 

Data Set 1: 

The following right to skewed dataset presented by Cordeiro and Brito (2012) [13] is obtained 

from the measurements on petroleum rock samples. The data consists of 48 rock samples from a 

petroleum reservoir. The dataset corresponds to twelve core samples from petroleum reservoirs 

that were sampled by four cross-sections. Each core sample was measured for permeability and 

each cross-section has the following variables: the total area of pores, the total perimeter of pores 

and shape. We analyze the shape perimeter by squared (area) variable and the observations are: 

0.0903296, 0.2036540, 0.2043140, 0.2808870, 0.1976530, 0.3286410,  

0.1486220, 0.1623940, 0.2627270, 0.1794550, 0.3266350, 0.2300810, 

0.1833120, 0.1509440, 0.2000710, 0.1918020, 0.1541920, 0.4641250,  

0.1170630, 0.1481410, 0.1448100,  0.1330830,  0.2760160,  0.4204770, 

0.1224170, 0.2285950, 0.1138520, 0.2252140, 0.1769690, 0.2007440,  

0.1670450,  0.2316230, 0.2910290,  0.3412730,  0.4387120,  0.2626510, 

0.1896510,  0.1725670,  0.2400770,  0.3116460,  0.1635860,  0.1824530,  

0.1641270,  0.1534810,  0.1618650,  0.2760160,  0.2538320,  0.2004470. 

Data set 2 

The symmetric behavior of the following dataset, discussed by Dasgupta (2011) [14], consists of 

50 observations relates to holes operation on jobs made of iron sheet. This dataset is as follows: 

0.04, 0.02, 0.06, 0.12, 0.14, 0.08, 0.22, 0.12, 0.08, 0.26, 0.24, 0.04, 0.14, 0.16, 0.08, 0.26, 0.32, 0.28, 0.14, 

0.16, 0.24, 0.22, 0.12, 0.18, 0.24, 0.32, 0.16, 0.14, 0.08, 0.16, 0.24, 0.16, 0.32, 0.18, 0.24, 0.22, 0.16, 0.12, 

0.24, 0.06, 0.02, 0.18, 0.22, 0.14, 0.06, 0.04, 0.14, 0.26, 0.18, 0.16. 



Int. J. Anal. Appl. 19 (3) (2021) 430 

 

 

Data Set 3: 

The following second data set presented by Cordeiro and Brito (2012) [13], displays the skewed 

symmetric trend of data. This data discusses the total milk production in the first birth of 107 

cows from SINDI race. These cows are property of the Carnaúba farm which belongs to the 

Agropecuária Manoel Dantas Ltda (AMDA), located in Taperoá City, Paraíba (Brazil). The data of 

proportion of total milk is as under: 

0.4365, 0.4260, 0.5140, 0.6907, 0.7471, 0.2605, 0.6196, 0.8781, 0.4990, 0.6058, 0.6891, 0.5770, 0.5394, 

0.1479, 0.2356, 0.6012, 0.1525, 0.5483, 0.6927, 0.7261, 0.3323, 0.0671, 0.2361, 0.4800, 0.5707, 0.7131, 

0.5853, 0.6768, 0.5350, 0.4151, 0.6789, 0.4576, 0.3259, 0.2303, 0.7687, 0.4371, 0.3383, 0.6114, 0.3480, 

0.4564, 0.7804, 0.3406, 0.4823, 0.5912, 0.5744, 0.5481, 0.1131, 0.7290, 0.0168, 0.5529, 0.4530, 0.3891, 

0.4752, 0.3134, 0.3175, 0.1167, 0.6750, 0.5113, 0.5447, 0.4143, 0.5627, 0.5150, 0.0776, 0.3945, 0.4553, 

0.4470, 0.5285, 0.5232, 0.6465, 0.0650, 0.8492, 0.8147, 0.3627, 0.3906, 0.4438, 0.4612, 0.3188, 0.2160, 

0.6707, 0.6220, 0.5629, 0.4675, 0.6844, 0.3413, 0.4332, 0.0854, 0.3821, 0.4694, 0.3635, 0.4111, 0.5349, 

0.3751, 0.1546, 0.4517, 0.2681, 0.4049, 0.5553, 0.5878, 0.4741, 0.3598, 0.7629, 0.5941, 0.6174, 0.6860, 

0.0609, 0.6488, 0.2747. 

 

Estimated Parameters by MLE with their S.E. and Goodness of Fit 

 Data set 1 

Model ̂   ̂   
̂   - ln( )L   KS AIC CAIC 

Power 

 

0.63 

(0.0909) 

  -6.0118 0.4295 14.0237 14.1107 

Beta 

( =1) 

 3.9643 

(0.5722) 

 -

30.2205 

0.9156 62.4412 62.5281 

Kumaraswamy 44.6597 

(17.5894) 

2.7187 

(0.2937) 

 -

52.4915 

0.1533 108.9831 109.2497 

Beta 

 

5.9417 

(1.1825) 

21.2057 

(4.3513) 

 -

55.6002 

0.1427 115.2004 115.4671 

Libby Novick 

Kumaraswamy 

5.5040 

(0.6078) 

0.7469 

(0.2042) 

10.4007 

(130.7012) 

-

57.7939 

0.0852 121.588 122.1334 

 



Int. J. Anal. Appl. 19 (3) (2021) 431 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 
 

Figure 6.1: pdf, cdf and Q-Q graphs of the densities for data set 1 



Int. J. Anal. Appl. 19 (3) (2021) 432 

 

 

Estimated Parameters by MLE with their S.E. and Goodness of Fit  

 Data set 2 

Model ̂   ̂   
̂   - ln( )L    KS AIC CAIC 

Power 

 

0.5033 

(0.0711) 

  -15.0139 0.4364 32.0279 32.1113 

Beta 

( =1) 

 5.4693 

(0.7735) 

 -44.0998 0.8786 90.1997 90.2831 

Kumaraswamy 

 

 33.1367  

(13.9342) 

2.0773 

(0.2551) 

 -56.0686 0.0902 116.1374 116.3927 

Beta 

 

2.6825 

(0.5074) 

13.8656 

(2.8295) 

 -54.6066 0.1214 113.2133 113.4686 

Libby Novick 

Kumaraswamy 

2.0376 

(0.2687) 

226.0664 

(980.0583) 

0.1342 

(0.5945) 

-56.2244 0.0900 118.4489 118.9707 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 6.2: pdf and cdf graphs of the compitators distributions for data set 2 

 



Int. J. Anal. Appl. 19 (3) (2021) 433 

 

 

 

Figure 6.3: Q-Q plots of the densities for data set 2 

 

Estimated Parameters by MLE with their S.E. and Goodness of Fit data3 

 

 

 

 

 

 

Model ̂   ̂   
̂   ln( )L−    KS AIC CAIC 

Power 

 

1.1123 

(0.1075) 

  -0.58545 0.2418 3.17090 3.209 

Beta 

( =1) 

 1.4181 

(0.1371) 

 -5.8304 0.9669 13.6608 13.6989 

Kumaraswamy 

 

3.4363  

(0.5822) 

2.1948 

(0.2224) 

 -25.3946 0.0769 54.7893 54.9047 

Beta 

 

2.4125  

(0.3145) 

2.8296 

(0.3745) 

 -23.7772 0.0816 51.5544 51.6698 

Libby Novick Kumaraswamy 1.7588 

(1.0411) 

18.3624 

(201.0970) 

0.1044 

(1.2976) 

-27.2446 0.0698 60.4892 60.7222 



Int. J. Anal. Appl. 19 (3) (2021) 434 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

Figure 6.4: pdf and cdf graphs of the compitators distributions for data set 3 

 

Figure 6.5: Q-Q plots of the densities for data set 3 



Int. J. Anal. Appl. 19 (3) (2021) 435 

 

 

6 CONCLUSION 

The LNK distribution has some properties like the LNB distribution but, there are numerous 

benefits of the LNK distribution over LNB distribution: the LNK distribution has simple closed 

form of both its cdf and quantile function and that’s why it is easy to use in simulation studies.The 

distribution and quantile function of LNK do not involve any special functions. To compare the 

proposed model with other models, we apply these models to three sets of real data from different 

fields of science and engineering and it is examined by using well-known statistics. We conclude 

that the LNK distribution is better than the power, Beta with (a=1), Beta distribution and 

Kumaraswamy distribution. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



Int. J. Anal. Appl. 19 (3) (2021) 436 

 

 

Appendix 1: Quantile values of LNK distribution 

 

a  b  c  0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 0.999 

0.5 0.5  0.1 0.49155 0.72089 0.83237 0.89632 0.93652 0.96296 0.98050 0.99171 0.99798 0.99998 1 

0.5 0.5 0.2 0.29135 0.54420 0.70361 0.80797 0.87890 0.92795 0.96158 0.98353 0.99597 0.99996 1 

0.5 0.5 0.6 0.07899 0.23413 0.40237 0.55900 0.69444 0.80539 0.89110 0.95181 0.98798 0.99988 0.99999 

0.5 0.5 1.0 0.03610 0.12960 0.26010 0.40960 0.56250 0.70560 0.82810 0.92160 0.98010 0.99980 0.99999 

0.5 0.5 1.5 0.01828 0.07438 0.16780 0.29416 0.44444 0.60493 0.75831 0.88581 0.97037 0.99970 0.99999 

0.5 0.5 3.0 0.00525 0.02493 0.06634 0.13845 0.25000 0.40495 0.59472 0.79012 0.94204 0.99940 0.99999 

0.5 0.5 6.0 0.00141 0.00734 0.02185 0.05224 0.11111 0.21777 0.39386 0.64000 0.88898 0.99880 0.99998 

0.5 0.5 10 0.00052 0.00283 0.00888 0.02278 0.05325 0.11851 0.25277 0.49827 0.82493 0.99800 0.99998 

1.0 1.0 0.1 0.52631 0.71428 0.81081 0.86956 0.90909 0.93750 0.95890 0.97561 0.98901 0.99899 0.99990 

1.0 1.0 0.2 0.35714 0.55555 0.68181 0.76923 0.83333 0.88235 0.92105 0.95238 0.97826 0.99798 0.99980 

1.0 1.0 0.6 0.15625 0.29411 0.41666 0.52631 0.62500 0.71428 0.79545 0.86956 0.93750 0.99397 0.99940 

1.0 1.0 1.0 0.10000 0.20000 0.30000 0.40000 0.50000 0.60000 0.70000 0.80000 0.90000 0.99000 0.99900 

1.0 1.0 1.5 0.06896 0.14285 0.22222 0.30769 0.40000 0.50000 0.60869 0.72727 0.85714 0.98507 0.99850 

1.0 1.0 3.0 0.03571 0.07692 0.12500 0.18181 0.25000 0.33333 0.43750 0.57142 0.75000 0.97058 0.99700 

1.0 1.0 6.0 0.01818 0.04000 0.06666 0.10000 0.14285 0.20000 0.28000 0.40000 0.60000 0.94285 0.99403 

1.0 1.0 10.0 0.01098 0.02439 0.04109 0.06250 0.0909 0.13043 0.18918 0.28571 0.47368 0.90825 0.99008 

2.0 2.0 0.1 0.59248 0.73576 0.81319 0.86269 0.89751 0.92368 0.94444 0.96184 0.97764 0.99449 0.998371 

2.0 2.0 0.2 0.46139 0.60921 0.70282 0.76984 0.82120 0.86253 0.89722 0.92775 0.95673 0.98907 0.99675 

2.0 2.0 0.6 0.28757 0.40544 0.49548 0.57148 0.63906 0.70143 0.76103 0.82049 0.88475 0.96824 0.99034 



Int. J. Anal. Appl. 19 (3) (2021) 437 

 

 

2.0 2.0 1.0 0.22653 0.32492 0.40415 0.47476 0.54119 0.60625 0.67251 0.74349 0.82690 0.94868 0.98406 

2.0 2.0 1.5 0.18656 0.27009 0.33935 0.40308 0.46517 0.52843 0.59585 0.67213 0.76838 0.92582 0.97637 

2.0 2.0 3.0 0.13308 0.19456 0.24718 0.29735 0.34831 0.40283 0.46458 0.54018 0.64719 0.86602 0.95434 

2.0 2.0 6.0 0.09452 0.13889 0.17751 0.21507 0.25412 0.29715 0.34781 0.41330 0.51469 0.77459 0.91442 

2.0 2.0 10.0 0.07334 0.10800 0.13838 0.16815 0.19943 0.23435 0.27618 0.33167 0.42164 0.68824 0.86823 

2.5 2.5 1.0 0.27941 0.37373 0.44615 0.50896 0.56705 0.62340 0.68063 0.74227 0.81622 0.93330 0.97426 

3.0 3.0 0.1 0.64095 0.75811 0.82327 0.86619 0.89717 0.9210 0.94038 0.95702 0.97268 0.99101 0.99632 

3.0 3.0 0.2 0.53323 0.65307 0.72871 0.78371 0.82677 0.86225 0.89283 0.92060 0.94813 0.98233 0.99270 

3.0 3.0 0.6 0.38309 0.48491 0.55810 0.61822 0.67111 0.71995 0.76713 0.81531 0.86978 0.95043 0.97871 

3.0 3.0 1.0 0.32557 0.41540 0.48216 0.53897 0.59088 0.64085 0.69143 0.74602 0.81222 0.92230 0.965489 

3.0 3.0 1.5 0.28551 0.36582 0.42659 0.47932 0.52858 0.57723 0.62799 0.68488 0.75764 0.89138 0.94991 

3.0 3.0 3.0 0.22749 0.29276 0.34308 0.38769 0.43040 0.47385 0.52089 0.57626 0.65255 0.81847 0.90856 

3.0 3.0 6.0 0.18092 0.23334 0.27416 0.31075 0.34627 0.38301 0.42366 0.47297 0.54441 0.72284 0.84343 

3.0 3.0 10.0 0.15271 0.19714 0.23187 0.26316 0.29369 0.32550 0.36102 0.40471 0.46950 0.64388 0.77952 

 

 

 

 

 

 

 

 

 



Int. J. Anal. Appl. 19 (3) (2021) 438 

 

 

Appendix 2: Variance, Skewness and Kurtosis of LNK distribution 

Following table shows the skewness and kutosis alongwith variance of LNK distribution 

a  b  c  Variance Skewness Kurtosis 

0.5 0.5 0.1 0.05296 −1.90608 6.01753 

0.5 0.5 0.2 0.07621 −1.27182 3.53694 

0.5 0.5 0.6 0.11146 −0.46476 1.80245 

0.5 0.5 1 0.12190 −0.13530 1.5386 

0.5 0.5 1.5 0.12589 0.11638 1.50603 

0.5 0.5 3.0 0.12346 0.54039 1.77633 

0.5 0.5 6.0 0.11145 0.97311 2.46619 

0.5 0.5 10.0 0.09890 1.30763 3.28243 

1.0 1.0 0.1 0.04264 -1.82849 5.97880 

1.0 1.0 0.2 0.05954 −1.19142 3.56560 

1.0 1.0 0.6 0.08049 −0.35626 1.95722 

1.0 1.0 1 0.08333 0.00000 1.80000 

1.0 1.0 1.5 0.08152 0.28209 1.89855 

1.0 1.0 3.0 0.07109 0.78491 2.56442 

1.0 1.0 6.0 0.05508 1.34776 4.06188 

1.0 1.0 10.0 0.04264 1.82849 5.97880 

1.5 1.5 0.1 0.03501 -87.1461 386.365 

1.5 1.5 0.2 0.04780 -40.6365 139.708 

1.5 1.5 0.6 0.06162 -13.6714 32.6900 

1.5 1.5 1 0.06205 −0.04546 2.00878 

1.5 1.5 1.5 0.05922 0.21997 2.1023 

1.5 1.5 3.0 0.04919 0.68276 2.73117 

1.5 1.5 6.0 0.03603 1.18148 4.11004 

1.5 1.5 10.0 0.02662 1.58988 5.82322 

2.0 2.0 0.1 0.02935 −1.80078 6.22199 

2.0 2.0 0.2 0.03939 −1.20368 3.91541 

2.0 2.0 0.6 0.04923 −0.43784 2.35643 

2.0 2.0 1 0.04888 −0.12530 2.18005 

2.0 2.0 3.0 0.03789 0.50497 2.69308 

2.0 2.0 6.0 0.02764 0.89112 3.64976 

2.0 2.0 10.0 0.02052 1.17422 4.71545 

2.5 2.5 0.1 0.02502 -694.098 3782.96 

2.5 2.5 0.2 0.03311 -273.261 1177.61 

2.5 2.5 0.6 0.04050 -278.916 868.895 

2.5 2.5 1.5 0.03761 0.00355 2.3443 

2.5 2.5 3.0 0.03094 0.33237 2.64818 

2.5 2.5 6.0 0.02295 0.62511 3.25548 

2.5 2.5 10.0 0.01744 0.81623 3.85719 

 



Int. J. Anal. Appl. 19 (3) (2021) 439 

 

 

Conflicts of Interest: The author declares that there are no conflicts of interest regarding the 

publication of this paper. 

 

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