Quartic Rank Transmuted Gumbel Distribution

Journal of Xidian University https://doi.org/10.37896/jxu16.6/023 ISSN No:1001-2400
VOLUME 16, ISSUE 6, 2022 http://xadzkjdx.cn/
Quartic Rank Transmuted Gumbel Distribution
Doaa ELhertaniy
PhD student at the Department of Mathematics,
university of the Holy Quran and Taseel of Science, Sudan
Email:dalhirtani@gmail.com
Abstract- In this paper, a Quartic Rank Transmuted Gumbel distribution (QTGD)
to an extended the work of Quartic transmuted distribution families. QTGD increases
the ability of the transmuted distributions to be flexible and facilitate the modelling of
more complex data. We study the main statistical properties of the Quartic transmuted
model, including its hazard rate function, moment-generating function, characteristic
function, quantile function, entropy, and order statistics. Finally, an application of QTGD
,using two real data sets are used to examine the ability to apply it and to observe the
performance of estimation techniques on a Quartic Rank Transmuted Gumbel(QTGD),
Cubic Transmuted Gumbel(CTGD), Transmuted Gumbel(TGD) and Gumbel(GD)
distributions. The observed results showed that QTGD gives better fit than CTGD,TGD,
and GD distributions for the applied data sets.
Keywords- Quartic rank transmuted, Gumbel Distribution , Rényi Entropy ,
Shannon Entropy
1 INTRODUCTION
The Gumbel distribution is named after Emil Julius Gumbel (1891–1966), based
on his original papers describing the distribution. The Gumbel distribution is a particular
case of the generalized extreme value distribution (also known as the Fisher-Tippett
distribution). It is also known as the log-Weibull distribution and the double exponential
distribution [9]. The Gumbel distribution is perhaps the most widely applied statistical
distribution for problems in engineering. It is also known as the extreme value
distribution of type I. Some of its recent application areas in engineering include flood
frequency analysis, network engineering, nuclear engineering, offshore engineering,
risk-based engineering, space engineering, software reliability engineering, structural
engineering, and wind engineering. A recent book by Kotz and Nadarajah [11], which
describes this distribution, lists over fifty applications ranging from accelerated life
testing through earthquakes, floods, horse racing, rainfall, queues in supermarkets, sea
currents, wind speeds, and track race records (to mention just a few).It is one of four
EVDs in common use. The other three are the Fréchet Distribution, the Weibull
Distribution, and the Generalized Extreme Value Distribution. The generalized extreme
value (GEV) distribution is a family of continuous probability distributions developed
within extreme value theory to combine the Gumbel, Fréchet, and Weibull families also
known as type I, II and III extreme value distributions. The probability density function
(PDF) and the cumulative distribution function (CDF) for Gumbel distribution are defined
as follow,

,
1
=
)
,
;
( w
we
X
g 


 (1)
where
.
0,
>
,
=
)
(
R
e
w
x







and
R
x
e
X
G w


;
=
)
,
;
( 
 (2)
Some extensions of the Gumbel distribution have previously been proposed. The
Beta Gumbel distribution,(Nadarajah et al. [13]), the Exponentiated Gumbel distribution
as a generalization of the standard Gumbel distribution introduced by Nadarajah
[12],and the Exponentiated Gumbel type-2 distribution, studied by Okorie et al. [14],
Transmuted Gumbel type-II distributton with applications in diverse fields of science by
Ahmad et al. [1], giving Transmuted exponentiated Gumbel distribution (TEGD) and its
application to water quality data of Deka et al. [7], and transmuted Gumbel distribution
(TGD) along with several mathematical properties has studied by Aryal and Tsokos [4]
using quadratic rank transmutation. Quadratic rank transmuted distribution has been
proposed by Shaw and Buckley [18]. A random variable X is said to have a quadratic rank
transmuted distribution if its cumulative distribution function is given by
1
|
|
,
)]
(
[
)
(
)
(1
=
)
( 2


 

 x
G
x
G
x
F (3)
Differentiating (3) with respect to x, it gives the probability density function (pdf)
of the quadratic rank transmuted distribution as
1
|
|
)],
(
2
)
)[(1
(
=
)
( 

 

 x
G
x
g
x
f (4)
where )
(x
G and )
(x
g are the cdf and pdf respectively of the base distribution. It is
very important observe that at  = 0 , we have the base original distribution. The
quadratic transmuted family of distributions given in (3) extends any baseline
distribution )
(x
G giving larger applicability. (Rahman et al. [15]) introduced the cubic
transmuted family of distributions. A random variable X is said to have cubic
transmuted distribution with parameter 1
 and 2
 if its cumulative distribution
function (cdf) is given by
3
2
2
1
2
1 )]
(
[
)]
(
)[
(
)
(
)
(1
=
)
( x
G
x
G
x
G
x
F 


 


 (5)
with corresponding pdf
R
x
x
G
x
G
x
g
x
f 



 )],
(
3
)
(
)
2(
)[1
(
=
)
( 2
2
1
2
1 


 (6)
where i
  [-1,1], i =1,2 are the transmutation parameters and obey the condition
1
2 2
1 


 
 . The proofs and the further details can be found in (Granzatto et al.
[8]).Recently, (Ali et al. [2]) introduced the Quartic transmuted family of distributions. A
random variable X is said to have Quartic transmuted distribution with parameter 1

, 2
 and 3
 if its cumulative distribution function (cdf) is given by

4
3
2
1
3
3
2
1
2
1
2
1 )]
(
)[
2
(1
)]
(
)[
2
2(
)]
(
)[
3(
)
(
2
=
)
( x
G
x
G
x
G
x
G
x
F 







 







 (7)
with corresponding pdf
R
x
x
G
x
G
x
G
x
g
x
f 








 )],
(
)
2
4(1
)
(
)
2
6(
)
(
)
6(
)[2
(
=
)
( 3
3
2
1
2
3
2
1
1
2
1 








(8)
where i
  [0,1], i =1,2.and 3
  [-1,1] are the transmutation parameters and
obey the condition 1
2 3
2 


 
 . This paper is organized as follows, the Quartic
transmuted Gumbel distribution is defined in Section 4. In Section 5 statistical
properties have been discussed, like shapes of the density and hazard rate functions,
quantile function, moments and moment-generating function, Characteristic Function ,
and cumulant generating function. Entropy in Section 6, for the Quartic transmuted
Gumbel distribution along with the distribution of order statistics in Section 7. Section 8
provides parameter estimation of Quartic transmuted Gumbel distribution. An
application of the QTGD to two real data sets for the purpose of illustration is conducted
in section 9. Finally, in Section 10, some conclusions are declared.
2 QUARTIC RANK TRANSMUTED GUMBEL Distribution
The quartic rank transmuted Gumbel distribution is defined as follows: The CDF
of a quartic rank transmuted Gumbel distribution is obtained by using (2) in (7)
]
)
2
(1
)
2
2(
)
3(
[2
=
)
( 3
3
2
1
2
3
2
1
1
2
1
w
w
w
w
e
e
e
e
x
F 











 







 (9)
where ,
R
x  , 0,
>
 i
  [0,1], i =1,2 and 3
  [-1,1] are the
transmutation parameters and obey the condition 1
2 3
2 


 
 .
It is very important note observe that at value
2
1
=
=
= 3
2
1 

 ,the quartic rank
transmuted Gumbel distribution reduce to Gumbel distribution according to the
transmutation map.
The probability density function (pdf) of a quartic rank transmuted Gumbel
distribution is given by
]
)
2
4(1
)
2
6(
)
6(
[2
1
=
)
( 3
3
2
1
2
3
2
1
1
2
1
w
w
w
w
e
e
e
we
x
f 











 









(10)
where
0
>
,
,
,
=
)
(




R
x
e
w
x



Figure 1 and 2, show the PDF and CDF of the quartic rank transmuted Gumbel
distribution for different values of parameters 1
 , 2
 and 3
 where  =3 and  =2.

Fig.1: The pdf of CTGD for different value of 1
 , 2
 and 3
 where  = 3 and = 2
Fig.2: The CDF of CTGD for different value of 1
 , 2
 and 3
 where  = 3 and =2

3 STATISTICAL PROPERTIES
3.1 Shapes of the density and hazard rate functions
The reliability function of the cdf F(x) of distribution is defined by
).
(
1
=
)
( x
F
x
R  For the quartic rank transmuted Gumbel (QTG) distribution, the
reliability function is given as,
]
)
2
(1
)
2
2(
)
3(
[2
1
=
)
( 3
3
2
1
2
3
2
1
1
2
1
w
w
w
w
e
e
e
e
X
R 












 








(11)
The hazard rate function can be written as the ratio of the pdf f (x) and the
reliability function R(x) = 1 - F(x). That is,
,
)
(
)
(
=
)
(
x
R
x
f
x
h
then we can find the hazard rate function of QTG distribution by (11) and (10):
]
)
2
(1
)
2
2(
)
3(
[2
1
]
)
2
4(1
)
2
6(
)
6(
[2
1
=
)
( 3
3
2
1
2
3
2
1
1
2
1
3
3
2
1
2
3
2
1
1
2
1
w
w
w
w
w
w
w
w
e
e
e
e
e
e
e
we
x
h 













































(12)
The cumulative hazard function is defined by
),
(
=
)
( x
lnR
x
H 
so the cumulative hazard function of the QTG distribution is
 .
]
)
2
(1
)
2
2(
)
3(
[2
1
=
)
( 3
3
2
1
2
3
2
1
1
2
1
w
w
w
w
e
e
e
e
ln
x
H 













 







 (13)
The reverse hazard function is
)
(
)
(
=
)
(
X
F
x
f
x
r (14)
Using (14), we can write the reverse hazard function of QTG distribution as
]
)
2
(1
)
2
2(
)
3(
[2
]
)
2
4(1
)
2
6(
)
6(
[2
=
)
( 3
3
2
1
2
3
2
1
1
2
1
3
3
2
1
2
3
2
1
1
2
1
w
w
w
w
w
w
e
e
e
e
e
e
w
x
r 










































(15)
The Odd function of a distribution with cdf )
(x
F is defined as
)
(
1
)
(
=
)
(
x
F
x
F
x
O

Then the Odd function of the QTG distribution is given as
 
 1
1
3
3
2
1
2
3
2
1
1
2
1 1
]
)
2
(1
)
2
2(
)
3(
[2
=
)
(















 w
w
w
w
e
e
e
e
x
O 







 (16)

Fig. 3: Reliability function of the QTGD for different value of 1
 , 2
 and 3
 where  = 3
and  =2
Fig. 4: Hazard function of the QTGD for different value of 1
 , 2
 and 3
 where  = 3
and  =2

3.2 Quantile Function
Here we will compute the quantile function of the quartic rank transmuted
Gumbel probability distribution.
Theorem 3.1 Let X be random variable from the quartic rank transmuted
Gumbel probability distribution with parameters 0,
>
 0,
>
 1
1 1 

  and
2
3
2 1 

 

 . Then the quantile function of X, is given by
))
,
,
,
(
log
(
log
= 3
2
1 



 q
B
xq 
 (17)
Proof: To compute the quantile function of the quartic rank transmuted Gumbel
probability distribution, we substitute x by q
x and )
(x
F by q in (9) to get the
equation
]
)
2
(1
)
2
2(
)
3(
[2
= )
)
(
3
3
2
1
)
)
(
2
3
2
1
)
)
(
1
2
1
)
)
(






































q
x
e
q
x
e
q
x
e
q
x
e
e
e
e
e
q
(18)
Then, we solve the equation (18) for .
q
x So, let .
= )
)
(
( 




q
x
e
e
y Thus, (18) becomes
4
3
2
1
3
3
2
1
2
1
2
1 )
2
(1
)
2
2(
)
3(
2
= y
y
y
y
q 







 








and hence,
0
=
2
)
3(
)
2
2(
)
2
(1 1
2
1
2
3
3
2
1
4
3
2
1 q
y
y
y
y 








 







 (19)
Let a = ),
2
(1 3
2
1 

 

 b = )
2
2( 3
2
1 

 
 , c = )
3( 1
2 
  , d = 1
2 and e=
-q then the equation (19) becomes
0.
=
2
3
4
e
dy
cy
by
ay 



Then, equation(20) from [6]
2
)
2
6
4(
)
4(
2
4
=
2
z
l
a
b
y










(20)
where
6
=
l
z 
 ,»» 2
2
8
3
=
a
b
a
c
l  ,»»
3
1
3
1
)
(
3
)
(
=


i
i
re
re
z  ,»»»»

 4
=
m

and,
a
d
a
b
m 
 2
3
16
=
Now, let the function B(q, 1
 , 2
 , 3
 ) be defined by

2
)
2
6
4(
)
4(
2
4
=
)
,
,
,
(
2
3
2
1
z
l
a
b
q
B













Hence,
)
,
,
,
(
=
= 3
2
1
)
)
(
(





q
B
e
y
q
x
e



(21)
Take natural Logarithm to both sides to get
)
,
,
,
(
log
= 3
2
1
)
(





q
B
e
q
x 


Then, we have the equation
))
,
,
,
(
log
(
log
= 3
2
1 



 q
B
xq 

The first quartile, median and third quartile can be obtained by setting q = 0.25, 0.50
and 0.75 in (17) respectively.
3.3 Moments and Moment-generating function
Moments function
Theorem 3.2 Let X  QTGD( , ). Then the rth moment of X is given by
))
(
(2
)
6(
)
(
[2
1)
(
=
)
(
=
)
( 1
2
1
0
=








 














 

 k
k
k
r
k
k
r
k
r
r k
k
k
r
x
E
x
E
1
=
|
))]
(
(4
)
2
4(1
))
(
(3
)
2
6( 3
2
1
3
2
1 









 











 

k
k
k
k
(22)
Proof: The rth moment of the positive random variable X with probability
density function is given by
dx
x
f
x
x
E r
r
)
(
=
)
(
0


(23)
Substituting from (10) in to (23),
dx
e
e
e
we
x
x
E w
w
w
w
r
r
]
)
2
4(1
)
2
6(
)
6(
[2
1
=
)
( 3
3
2
1
2
3
2
1
1
2
1
0














 









(24)
where
)
(
= 



x
e
w , Then dx
e
dw
x



)
(
=



, »» lnw
x 
 
=
With substitution in (24) using Gamma integration;
))
(
(2
)
6(
)
(
[2
1)
(
=
)
( 1
2
1
0
=








 














 

 k
k
k
r
k
k
r
k
r k
k
k
r
x
E

1
=
|
))]
(
(4
)
2
4(1
))
(
(3
)
2
6( 3
2
1
3
2
1 









 











 

k
k
k
k
Moment Generating Function
Theorem 3.3 The moment generating function )
(t
Mx of a random variable X
QTGD (  , ) is given by
))
(
(2
)
6(
)
(
[2
1)
(
!
=
)
( 1
2
1
0
=
0
=








 














 



 k
k
k
r
k
k
r
k
r
r
x
k
k
k
r
r
t
t
M
1
=
|
))]
(
(4
)
2
4(1
))
(
(3
)
2
6( 3
2
1
3
2
1 









 











 

k
k
k
k
(25)
Proof:The moment generating function of the positive random variable X
with probability density function is given by
dx
x
f
e
e
E
t
M tx
tx
x )
(
=
)
(
=
)
(
0


Using series expansion of tx
e ,
)
(
!
=
)
(
!
=
)
(
0
=
0
0
=
r
r
n
r
r
r
n
r
x x
E
r
t
dx
x
f
x
r
t
t
M 



Then
))
(
(2
)
6(
)
(
[2
1)
(
!
=
)
( 1
2
1
0
=
0
=








 














 



 k
k
k
r
k
k
r
k
r
r
x
k
k
k
r
r
t
t
M
1
=
|
))]
(
(4
)
2
4(1
))
(
(3
)
2
6( 3
2
1
3
2
1 









 











 

k
k
k
k
3.4 Characteristic Function
The characteristic function theorem of the quartic transmuted Gumbel
distribution is stated as follow,
Theorem 3.4 Suppose that the random variable X have the QTGD (  , ), then
characteristic function, )
(t
x
 ,is
))
(
(2
)
6(
)
(
[2
1)
(
!
)
(
=
)
( 1
2
1
0
=
0
=









 














 



 k
k
k
r
k
k
r
k
r
r
x
k
k
k
r
r
it
t
1
=
|
))]
(
(4
)
2
4(1
))
(
(3
)
2
6( 3
2
1
3
2
1 









 











 

k
k
k
k
(26)
Where 1
= 
i and t  
3.5 Cumulant Generating Function
The cumulant generating function is defined by

)
(
log
=
)
( t
M
t
K x
e
x
Cumulant generating function of quartic rank transmuted Gumbel distribution is
given by
))
(
(2
)
6(
)
(
[2
1)
(
!
log
=
)
( 1
2
1
0
=
0
=








 














 



 k
k
k
r
k
k
r
k
r
r
e
x
k
k
k
r
r
t
t
K
1
=
|
))]
(
(4
)
2
4(1
))
(
(3
)
2
6( 3
2
1
3
2
1 









 











 

k
k
k
k
(27)
4 ENTROPY
4.1 Rényi Entropy
If X is a non-negative continuous random variable with pdf ),
(x
f then the
Rényi entropy of order  (See Rényi [16]) of X is defined as
1)
(
0,
>
,
)]
(
[
1
1
=
)
(
0


 





 dx
x
f
log
x
H (28)
Theorem 4.1 The Rényi entropy of a random variable X  QTGD ( , ),
with 
1
 1, 
2
 0 , 
3
 0 and 3
2
1 

 
 is given by
r
k
j
j
k
r
j
k
j
r
k
j
c
x
H )
2
(1
)
(
1
)
,
,
,
(
[
log
1
1
=
)
( 3
2
1
1
2
1
1
0
=
0
=
0
=











 












]
)
(
)
(
)
2
( 3
2
1 





r
j
k
r
k






 
Proof: Suppose X has the pdf in (10). Then, one can calculate
 














]
)
2
4(1
)
2
6(
)
6(
[2
1
=
)]
(
[ 3
3
2
1
2
3
2
1
1
2
1
w
w
w
w
e
e
e
e
w
x
f 











 (29)
By the general binomial expansion, we have
 








 ]
)
2
4(1
)
2
6(
)
6(
[2 3
3
2
1
2
3
2
1
1
2
1
w
w
w
e
e
e 











 j
w
w
w
j
j
e
e
e
j
3
3
2
1
2
3
2
1
1
2
1
0
=
)
2
4(1
)
2
6(
)
6(
)
(2
= 




















 








 
(30)
by the Binomial Theorem,
 j
w
w
w
e
e
e 3
3
2
1
2
3
2
1
1
2 )
2
4(1
)
2
6(
)
6( 









 







   k
w
w
k
j
w
j
k
e
e
e
k
j 3
3
2
1
2
3
2
1
1
2
0
=
)
2
4(1
)
2
6(
)
6(
= 


















 






 (31)

by the Binomial Theorem,
 k
w
w
e
e 3
3
2
1
2
3
2
1 )
2
4(1
)
2
6( 






 





   r
w
r
k
w
k
r
e
e
r
k 3
3
2
1
2
3
2
1
0
=
)
2
4(1
)
2
6(
= 















 





rw
r
r
w
r
k
r
k
r
k
k
r
e
e
r
k 3
3
2
1
)
2(
3
2
1
0
=
)
2
(1
4
)
2
(
6
= 

















 





w
r
k
r
k
r
r
r
k
k
r
e
r
k )
(2
3
2
1
3
2
1
0
=
)
2
(
)
2
(1
4
6
= 
















 




 (32)
Substitute from(32) in (31), to get
 j
w
w
w
e
e
e 3
3
2
1
2
3
2
1
1
2 )
2
4(1
)
2
6(
)
6( 









 







  


























 






 w
r
k
r
k
r
r
r
k
k
r
k
j
w
j
k
e
r
k
e
k
j )
(2
3
2
1
3
2
1
0
=
1
2
0
=
)
2
(
)
2
(1
4
6
)
6(
= 







w
r
j
k
r
k
r
k
j
r
r
j
k
r
j
k
e
r
k
k
j )
(
3
2
1
3
2
1
1
2
0
=
0
=
)
2
(
)
2
(1
)
(
(4)
6
= 




























 






 (33)
Now, substitute from(33) in (30), to get
 








 ]
)
2
4(1
)
2
6(
)
6(
[2 3
3
2
1
2
3
2
1
1
2
1
w
w
w
e
e
e 











r
k
j
r
r
j
k
r
j
k
j
j r
k
k
j
j
)
2
(1
)
(
(4)
6
[
)
(2
= 3
2
1
1
2
0
=
0
=
1
0
=






 



























 






]
)
2
( )
(
3
2
1
w
r
j
k
r
k
e 





 


w
r
j
k
r
k
r
k
j
j
r
r
j
j
k
r
j
k
j
e
r
k
k
j
j
)
(
3
2
1
3
2
1
1
2
1
0
=
0
=
0
=
)
2
(
)
2
(1
)
(
(4)
6
2
= 








































 








 

(34)
Now, substitute from(34) in (29), to get
k
j
j
r
r
j
j
k
r
j
k
j
w
r
k
k
j
j
e
w
x
f 
































 )
(
(4)
6
2
1
=
)]
(
[ 1
2
1
0
=
0
=
0
=











w
r
j
k
r
k
r
e )
(
3
2
1
3
2
1 )
2
(
)
2
(1 








 






Let c( j, k, r , )= r
r
j
j
r
k
k
j
j
(4)
6
2 
























 

, then
r
k
j
j
k
r
j
k
j
r
k
j
c
x
f )
2
(1
)
(
)
,
,
,
(
1
=
)]
(
[ 3
2
1
1
2
1
0
=
0
=
0
=














 





)
)
(
(
)
(
3
2
1 )
2
( 

















x
w
r
j
k
r
k
e (35)
To find )
(x
H , we substitute from (35) in (28)
r
k
r
k
j
j
k
r
j
k
j
r
k
j
c
log
x
H 












 )
2
(
)
2
(1
)
(
)
,
,
,
(
1
[
1
1
=
)
( 3
2
1
3
2
1
1
2
1
0
=
0
=
0
=















].
)
)
(
(
)
(
0
dx
e
x
w
r
j
k













 (36)
We can evaluate the integration
dx
e
e
dx
e w
r
j
k
x
x
w
r
j
k
)
(
)
)
(
(
0
)
)
(
(
)
(
0
)
(
= 

























(37)
where
)
(
= 



x
e
w , and 1

 , then dx
e
dw
x



)
(
=



and 
<
<
0 w
With substitution with this transformation in (37) using Gamma integration,







)
(
)
(
=
)
(
1
0 r
j
k
dw
e
w w
r
j
k





 





 (38)
After solving the integral, we get the Rényi entropy of the QTGD (  , ) by substitute
from (38) in (36)
r
k
j
j
k
r
j
k
j
r
k
j
c
x
H )
2
(1
)
(
1
)
,
,
,
(
[
log
1
1
=
)
( 3
2
1
1
2
1
1
0
=
0
=
0
=











 












]
)
(
)
(
)
2
( 3
2
1 





r
j
k
r
k






 
4.2 q-Entropy
The q-entropy was introduced by (Havrda and Charvat [10]). It is the one
parameter generalization of the Shannon entropy.( Ullah [20]) defined the q-entropy as
1
0,
>
,
)
(
1
1
1
=
)
(
0






 
 

andq
whereq
dx
x
f
q
q
I q
H (39)
Theorem 4.2 The q-entropy of a random variable X  QTGD (  , ), with


1
 1, 
2
 0, 
3
 0 and 3
2
1 

 
 . is given by
r
k
j
j
q
q
k
r
j
k
j
H q
r
k
j
c
q
q
I )
2
(1
)
(
1
)
,
,
,
(
[1
1
1
=
)
( 3
2
1
1
2
1
1
0
=
0
=
0
=





















]
)
(
)
(
)
2
( 3
2
1 q
r
k
r
j
k
q
q






 



Proof. To find )
(q
IH , we substitute (35) in (40).
4.3 Shannon Entropy
The (Shannons entropy [17]) of a non-negative continuous random variable X
with pdf )
(x
f is defined as
dx
x
f
x
f
x
f
E
f
H ))
(
(
log
)
(
=
)]
(
log
[
=
)
(
0



 (40)
Below, we are going to use the Expansion of the Logarithm function (Taylor series at 1),
1
|<
|
,
1)
(
1)
(
=
)
(
log 1
0
=
x
m
x
x
m
m
m

 

 (41)
The Shannon entropy of a random variable X  QTGD ( , ), with 
1
 1, 
2

0, 
3
 0 and 3
2
1 

 
 . is given by
k
j
j
n
n
n
k
r
j
k
n
j
m
n
m
n
r
k
j
c
m
n
m
f
H 




















 )
(
1
1)
,
,
,
(
1
1)
(
=
)
( 1
2
1
1
1
0
=
0
=
1
0
=
0
=
1
=




(42)
1
3
2
1
3
2
1
)
1
(
1)
(
)
2
(
)
2
(1 












 n
r
k
r
r
k
j
n
n





 (43)
Proof: By the Expansion of the Logarithm function (41)
,
1)
)
(
(
1)
(
=
))
(
(
log 1
1
= m
x
f
x
f
m
m
m

 


and by Binomial Theorem,















 



 )
(
1)
(
1
1)
(
=
0
=
1
1
=
x
f
n
m
m
n
n
m
m
n
m
m
)
(
1
1)
(
=
))
(
(
log 1
0
=
1
=
x
f
m
n
m
x
f n
n
m
n
m








 


 (44)
To compute the Shannon’s entropy of X, substitute from (44) in (40)
dx
x
f
m
n
m
x
f
dx
x
f
x
f
f
H n
n
m
n
m
)
(
1
1)
(
)
(
=
))
(
(
log
)
(
=
)
( 1
0
=
1
=
0
0 









 







dx
x
f
m
n
m
f
H n
n
m
n
m
)
(
1
1)
(
=
)
( 1
0
=
1
=
0














 (45)

substituting from (35) in (45), to get
k
j
j
n
n
k
r
j
k
n
j
n
m
n
m
n
r
k
j
c
m
n
m
f
H 
















 




 )
(
1
1)
,
,
,
(
1
1)
(
=
)
( 1
2
1
1
1
0
=
0
=
1
0
=
0
=
1
=
0




dx
e
x
n
w
r
j
k
n
r
k
r
)
)
(
1)
(
(
)
1
(
3
2
1
3
2
1 )
2
(
)
2
(1 























k
j
j
n
n
n
k
r
j
k
n
j
m
n
m
n
r
k
j
c
m
n
m
f
H 




















 )
(
1
1)
,
,
,
(
1
1)
(
=
)
( 1
2
1
1
1
0
=
0
=
1
0
=
0
=
1
=




dx
e
x
n
w
r
j
k
n
r
k
r
)
)
(
1)
(
(
)
1
(
0
3
2
1
3
2
1 )
2
(
)
2
(1 

























Now, we use (38) to fined the value of the integration, so
k
j
j
n
n
n
k
r
j
k
n
j
m
n
m
n
r
k
j
c
m
n
m
f
H 




















 )
(
1
1)
,
,
,
(
1
1)
(
=
)
( 1
2
1
1
1
0
=
0
=
1
0
=
0
=
1
=




1
3
2
1
3
2
1
)
1
(
1)
(
)
2
(
)
2
(1 












 n
r
k
r
r
k
j
n
n






5 ORDER STATISTICS
Let n
X
X
X ,...,
, 2
1 be a random sample of size k from the QTG distribution with
parameters 0
>
 , 0
>
 , 1
0 1 
  , 1
2
1 1 

 

 and 2
3
2 1 

 

 From [5],
the pdf of the k th order statistics is obtain by
k
n
k
k
X x
F
x
F
x
f
k
n
k
x
f 










)]
(
[1
)]
(
)[
(
=
)
( 1
)
(
(46)
Let k
X be the k th order statistic from X  QTGD ( , ) with 0
1 
 , 0
2 

and 0
3 
 . Then, the pdf of the k th order statistic is given by
Theorem,
byBinomial
,
)]
(
[1
)]
(
)[
(
=
)
( 1
)
(
k
n
k
k
X x
F
x
F
x
f
k
n
k
x
f 























 











 j
j
k
n
j
k
x
F
j
k
n
x
F
x
f
k
n
k )]
(
[
1)
(
)]
(
)[
(
=
0
=
1
1
0
=
)]
(
)[
(
1)
(
= 









 









 j
k
j
k
n
j
x
F
x
f
j
k
n
k
n
k
w
w
j
k
n
j
k
X e
we
j
k
n
k
n
k
x
f 











 









 )
6(
[2
1
1)
(
=
)
( 1
2
1
0
=
)
(




]
)
2
4(1
)
2
6( 3
3
2
1
2
3
2
1
w
w
e
e 







 






  1
3
3
2
1
2
3
2
1
1
2
1 ]
)
2
(1
)
2
2(
)
3(
[2
















j
k
w
w
w
w
e
e
e
e 








Then, the pdf of first order statistic (1)
X of QTG distribution is given as
]
)
2
4(1
)
2
6(
)
6(
[2
1
=
)
( 3
3
2
1
2
3
2
1
1
2
1
(1)
w
w
w
w
X e
e
e
we
n
x
f 











 









 
  1
3
3
2
1
2
3
2
1
1
2
1 ]
)
2
(1
)
2
2(
)
3(
[2
1
















n
w
w
w
w
e
e
e
e 








Therefore, the of the largest order statistic )
(n
X of QTG distribution is given by
w
w
w
w
n
X e
e
e
we
n
x
f 3
3
2
1
2
3
2
1
1
2
1
)
(
)
2
4(1
)
2
6(
)
6(
[2
1
=
)
( 











 









  1
3
3
2
1
2
3
2
1
1
2
1 )
2
(1
)
2
2(
)
3(
[2















n
w
w
w
w
e
e
e
e 








6 MAXIMUM LIKELIHOOD ESTIMATION (MLE)
Assume n
X
X
X ,...,
, 2
1 be a random sample of size n from QTGD(  , ) then
the likelihood function can be written as
i
w
i
w
i
n
i
i
n
i
e
e
w
x
f
l





 )
6(
[2
1
[
=
)
(
=
)
,
,
,
,
( 1
2
1
1
=
1
=
3
2
1 








]]
)
2
4(1
)
2
6(
3
3
2
1
2
3
2
1
i
w
i
w
e
e








 





where
)
(
= 


 i
x
i e
w , n
i 1,2,...,
= .
Then
i
w
n
i
i
w
i
n
i
n
e
e
w
l





 )
6(
[[2
1
=
)
,
,
,
,
( 1
2
1
1
=
1
=
3
2
1 







 (47)
]]
)
2
4(1
)
2
6(
3
3
2
1
2
3
2
1
i
w
i
w
e
e








 





Then, the Log likelihood function of a vector of parameters given as,
i
i
n
i
i
n
i
w
x
n
x
f
l 



 
 






 [
log
=
)
(
log
=
)
,
,
,
,
(
log
1
=
1
=
3
2
1
]]
)
2
4(1
)
2
6(
)
6(
[2
log
3
3
2
1
2
3
2
1
1
2
1
i
w
i
w
i
w
e
e
e












 








Differentiate w.r.t parameters  ,  , 1
 , 2
 and 3
 we have
i
n
i
w
n
l




1
=
1
=
log



(48)

i
w
i
w
i
w
i
w
i
w
i
i
w
i
n
i e
e
e
e
e
w
e
w
3
3
2
1
2
3
2
1
1
2
1
3
3
2
1
2
3
2
1
1
2
1
= )
2
2(1
)
2
3(
)
3(
)
2
6(1
)
2
6(
)
3(
1






















  

















2
2
1
=
=
log













 i
i
i
n
i
x
w
x
n
l
(49)
)
)
2
2(1
)
2
3(
)
3(
(
]
)
2
6(1
)
2
6(
)
)[3(
(
3
3
2
1
2
3
2
1
1
2
1
2
3
3
2
1
2
3
2
1
1
2
1
=
i
w
i
w
i
w
i
w
i
w
i
i
w
i
i
n
i e
e
e
e
e
w
e
w
x























  


















i
w
i
w
i
w
i
w
i
w
i
w
n
i e
e
e
e
e
e
l
3
3
2
1
2
3
2
1
1
2
1
3
2
1
=
1 )
2
2(1
)
2
3(
)
3(
2
3
3
1
=
log




















 









(50)
i
w
i
w
i
w
i
w
i
w
i
w
n
i e
e
e
e
e
e
l
3
3
2
1
2
3
2
1
1
2
1
3
2
1
=
2 )
2
2(1
)
2
3(
)
3(
2
6
3
=
log



















 









(51)
i
w
i
w
i
w
i
w
i
w
n
i e
e
e
e
e
l
3
3
2
1
2
3
2
1
1
2
1
3
2
1
=
3 )
2
2(1
)
2
3(
)
3(
4
3
=
log

















 









(52)
We can obtain the estimates of the unknown parameters by the maximum
likelihood method by setting these above nonlinear equations(48),(49),(50),(51)and(52)
to zero and solving them simultaneously. Therefore, statistical software can be
employed in obtaining the numerical solution to the nonlinear equations,We can
compute the maximum likelihood estimators (MLEs) of parameters (  ,  , 1
 ,
2
 ,and 3
 ) using quasi-Newton procedure,or computer packages softwares such as R,
SAS, MATLAB, MAPLE and MATHEMATICA.
7 APPLICATIONS
In this section, the Quartic Rank Transmuted Gumble Distribution (QTGD) is
applied on two data sets as follows:
Data Set 1: The values of data about strengths of 1.5 cm glass fibers [19, 3]. The
data set is: 0.55, 0.74, 0.77, 0.81, 0.84, 0.93, 1.04, 1.11, 1.13, 1.24, 1.25, 1.27, 1.28, 1.29,
1.30, 1.36, 1.39, 1.42, 1.48, 1.48, 1.49, 1.49, 1.50, 1.50, 1.51, 1.52, 1.53, 1.54, 1.55, 1.55,
1.58, 1.59, 1.60, 1.61, 1.61, 1.61, 1.61, 1.62, 1.62, 1.63, 1.64, 1.66, 1.66, 1.66, 1.67, 1.68,
1.68, 1.69, 1.70, 1.70, 1.73, 1.76, 1.76, 1.77, 1.78, 1.81, 1.82, 1.84, 1.84, 1.89, 2.00, 2.01,
2.24.
Data Set 2: The data represents a wind velocity (WVD) involving 264
observations of the maximum of monthly wind speed (mph) in Falconara Marittima,

Ancona, Italy for the months January 2000 to December, 2021. Data is available for
download from Weather underground(https://www.wunderground.com/) website.The
data set is: 24, 21, 28, 20, 20, 20, 24, 20, 39, 25, 26, 22, 33, 25, 28, 22, 20, 26, 21, 26, 22,
13, 21, 28, 25, 23, 30, 20, 18, 25, 17, 28, 16, 25, 26, 24, 21, 25, 23, 28, 15, 16, 20, 22, 22,
35, 21, 28, 2, 26, 18, 22, 26, 18, 36, 31, 40, 28, 31, 26, 32, 26, 29, 26, 23, 29, 26, 25, 29,
17, 24, 28, 35, 31, 38, 18, 26, 23, 18, 31, 21, 18, 28, 15, 23, 33, 28, 25, 22, 22, 26, 22, 75,
106, 22, 48, 99, 26, 105, 62, 22, 75, 78, 23, 86, 30, 25, 86, 21, 30, 29, 23, 100, 23, 29, 39,
21, 33, 38, 25, 22, 29, 25, 29, 23, 24, 22, 101, 22, 20, 20, 52, 21, 32, 23, 36, 24, 21, 21,
61, 23, 26, 16, 33, 25, 26, 21, 28, 23, 20, 21, 25, 24, 33, 18, 29, 20, 26, 33, 20, 24, 16, 21,
23, 76, 20, 38, 17, 29, 23, 25, 20, 23, 21, 23, 22, 36, 28, 16, 35, 64, 32, 35, 56, 72, 40, 24,
18, 26, 40, 24, 16, 29, 35, 29, 21, 20, 28, 25, 26, 22, 18, 29, 25, 36, 24, 29, 29, 28, 22, 26,
28, 29, 29, 26, 58, 29, 23, 92, 40, 18, 20, 23, 20, 28, 23, 26, 25, 25, 31, 31, 20, 25, 18, 33,
21, 25, 31, 26, 26, 21, 36, 24, 18, 32, 18, 21, 22, 24, 24, 22, 21, 32, 22, 26, 28, 25, 22, 23,
25, 16, 26, 24, 29
Summary statistics of the two data sets is demonstrated in Table 1. For the sake
of comparison and using these data sets, four alternative distributions:(QTGD) model,
the Gumbel distribution (GD) , transmuted Gumbel distribution (TGD) , Cubic
Transmuted Gumbel (CTGD) have been compared. The estimated values of the model
parameters along with corresponding standard errors are presented in Tables 2 and 4
for selected models using the MLE method.
In Tables 3 and 5, the goodness of fit of the Quartic Rank Transmuted Gumble
Distribution (QTGD)model,Cubic Transmuted Gumbel Distribution (CTGD), Transmuted
Gumbel Distribution (TGD) and the Gumbel distribution (GD) has been introduced using
different comparison measures we consider some criteria like (-2L )
( ): where is
L )
( the maximum value of log-likelihood function, AIC (Akaike Information
Criterion), AICc (Corrected Akaike Information Criterion) and BIC (Bayesian Information
Criterion),HQIC (Hanan-Quinn information criterion) and CAIC (Consistent Akaike
information criterion), for the data set. In general the better fit of the distribution
corresponds to the smaller value of the statistics (-2L )
( ) , AIC, CAIC, BIC, HQIC and
AICc.
Plots of the empirical and theoretical cdfs and pdfs for fitted distributions are
given in Figure 5, and Figure 6, respectively. These Figures shows that: the curve of the
pdf and cdf QTGD is closer to the curve of the sample of data than the curve of the pdf
and cdf of CTGD,TGD and GD . So, the QTGD is a better model than one based on the
CTGD,TGD and GD.
Table 1: Descriptive Statistics of Data Set 1, 2
Data mean Median Skewness kurtosis
Set1 1.59 1.506825 0.89993 3.92376
set2 29.01 25 3.0962617 13.3250012

Table 2: MLE’s of the parameters and respective SE’s for various distributions for Data
Set 1
Distribution Parameter Estimate SE
QTGD
 1.02455 0.00359295
 0.258847 0.000343268
1
 0.241234 0.0425602
2
 0.208044 0.287866
3
 -0.885227 0.574946
CTGD
 1.11008 0.00244777
 0.282105 0.000685827
1
 -0.305918 0.103321
2
 -1.69408 0.75491
TGD
 1.5819 -0.00316457
 0.488214 0.000263098
 0.999999 -0.177738
GD
 1.33313 0.00255327
 0.376428 0.00104844
Table 3: Goodness-of fit statistics using the selection criteria values for Data Set 1
ModeI -2L )
( AIC BIC HQIC AICc CAIC
QTGD 34.935 44.935 55.651 42.042 45.988 60.651
CTGD 41.6642 49.6642 58.237 47.350 50.354 62.237
TGD 50.701 56.701 63.130 54.965 57.108 66.130
GD 61.0358 65.058 69.322 63.879 65.258 71.322
(a) (b)
Fig.5 (a):Fitted pdf for Data Set 1 (b) Empirical cdf and theoretical cdf for Data Set 1

Table 4: MLE’s of the parameters and respective SE’s for various distributions for Data
Set 2
Distribution Parameter Estimate SE
QTGD
 28.573 -0.198022
 10.1077 0.412069
1
 1 -0.0374996
2
 1 0.0048907
3
 -0.38496 0.060622
CTGD
 27.5443 -0.522687
 9.2936 0.140624
1
 1 -0.0929894
2
 -0.444086 -0.0911491
TGD
 26.6412 0.374464
 9.15995 0.235455
 0.677728 0.00668499
GD
 23.7077 0.260879
 8.02217 0.148993
Table 5: Goodness-of fit statistics using the selection criteria values for Data Set 3
ModeI -2L )
( AIC BIC HQIC AICc CAIC
QTGD 1898.234 1908.234 1926.114 1906.826 1908.467 1931.114
CTGD 1939.172 1947.172 1961.476 1946.046 1974.326 1965.476
TGD 1951.112 1957.112 1967.839 1956.267 1957.204 1970.839
GD 1976.482 1980.482 1978.634 1979.919 1980.528 1989.634
(a) (b)
Fig.6 (a):Fitted pdf for Data Set 2 (b) Empirical cdf and theoretical cdf for Data Set 2.

8. CONCLUSION
In this paper, the Quartic Rank Transmuted Gumbel Distribution has been
introduced. This distribution is high elastic to treat the complex data.The graphical
representations of pdf, cdf, reliability function, hazard function are given for various
value of the parameters.We have discussed some statistical properties of the Quartic
Rank Transmuted Gumbel Distribution (QTGD) are discussed including moments,
moment generating function, characteristic function, quantile function, reliability
function, hazard function ,Rényi Entropy, q-Entropy , Shannon entropy and order
statistics. The model parameters are estimated by the maximum likelihood method. The
Quartic Rank Transmuted Gumbel Distribution has been applied on two real applications
and the obtained results showed that the Quartic Rank Transmuted Gumbel distribution
offers better appropriate than Gumbel , transmuted Gumbel and cubic transmuted
Gumbel distributions for the applied data sets.
References
[1] Ahmad, A.; ul Ain, S.Q.; Tripathi, R.: Transmuted gumbel type-II distribution with
applications in diverse fields of science. Pak. J. Statist, 37:429–446 (2021)
[2] Ali, M.; Athar, H.: Generalized rank mapped transmuted distribution for generating
families of continuous distributions.J. Statist. Theory and Appl. 20(1),132–248 (2021).
[3] Arifa, S.; Yab, M. Z.; Ali, A.: The modified burr iii g family of distributions. J. Data
Science 15(1),41–60 (2017)
[4] Aryal, G. R. ; Tsokos, C. P.: On the transmuted extreme value distribution with
application. Nonlinear Analysis Theory, Methods and Appl., 71:e1401–e1407 (2009)
[5] Casella, G. ; Berger, R. L.: Statist. inference. Cengage Learning (2021)
[6] Das, A.: A novel method to solve cubic and quartic equations. Preprint J. (2014)
[7] Deka,D.; Das, B.; Baruah, B. K.: Transmuted exponentiated gumbel distribution
(tegd) and its application to water quality data. Pak. J. of Statist. and Operation
Research, 115–126 (2017).
[8] Granzotto,D. C. T.; Louzada, F.; Balakrishnan, N.: Cubic rank transmuted
distributions: inferential issues and applications. J. Statist. Computation and Simulation
87,2760–2778 (2017).
[9] Gumbel, E. J.: Les valeurs extremes des distributions statistiques,vol. 5, 115–158
(1935).
[10] Havrda, J.; Charv´at, F.: Quantification method of classification processes. concept
of structural a-entropy. Kybernetika J. 3(1),30–35 (1967).
[11] Kotz, S.; Nadarajah, S.: Extreme value distributions: theory and applications.
World Scientific (2000).
[12] Nadarajah, S.: The exponentiated gumbel distribution with climate application.
Environmetrics, The official j. of the International Environmetrics Soc. 17,13–23 (2006)
[13] Nadarajah, S.; Kotz, S.: The beta gumbel distribution. Mathematical Problems in
engineering 323–332 (2004).

[14] Okorie, I. E.; Akpanta, A. C.; Ohakwe, J.: The exponentiated gumbel type-2
distribution: properties and application. International J. Mathematics and Mathematical
Sciences, (2016).
[15] Rahman, M. M.; Al-Zahrani, B.; Shahbaz, M. Q. : A general transmuted family of
distributions. Pak. J. of Statist. and Operation Research. 451–469 (2018).
[16] Rényi, A.: On measures of entropy and information.In Proceedings of the Fourth
Berkeley Symposium on Mathematical Statistics and Probability, Vo.4: Contributions to
the Theory of Statistics.pp.547–562.University of California Press(1961).
[17] Shannon, C. E.: A mathematical theory of communication. The Bell system technical
j. 27(3),379–423 (1948).
[18] Shaw, W. T.; Buckley, I. R. C.: The alchemy of probability distributions: beyond
gram-charlier expansions, and a skew-kurtotic-normal distribution from a rank
transmutation map. arXiv preprint arXiv:0901.0434 (2009).
[19] Smith, R. L.; Naylor, J.: A comparison of maximum likelihood and Bayesian
estimators for the three-parameter Weibull distribution. J. Royal Statist.Soc.: Series C
(Appl. Statis.), 36(3),358–369 (1987).
[20] Ullah, A.: Entropy, divergence and distance measures with econometric
applications. J. Statist. Planning and Inference, 49(1), 137-162 (1996).

Quartic Rank Transmuted Gumbel Distribution

Recommended

Recommended

More Related Content

Similar to Quartic Rank Transmuted Gumbel Distribution

Similar to Quartic Rank Transmuted Gumbel Distribution (20)

Recently uploaded

Recently uploaded (20)

Quartic Rank Transmuted Gumbel Distribution