Journal of Xidian University
https://doi.org/10.37896/jxu16.6/023
ISSN No:1001-2400
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,
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g ( X ; , ) =
1

ISSN No:1001-2400
we  w ,
(1)
where
w=e
(
x

)
,  > 0,   R.
and
G( X ;  ,  ) = e  w ; x  R
(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
F ( x) = (1   )G( x)  [G( x)]2 , |  | 1
Differentiating (3) with respect to x, it gives the probability density function (pdf)
of the quadratic rank transmuted distribution as
f ( x) = g ( x)[(1   )  2G( x)], |  | 1
where G(x) and g (x) 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 G(x) 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
F ( x) = (1  1 )G( x)  (2  1 )[G( x)]2  2 [G( x)]3
with corresponding pdf
f ( x) = g ( x)[1  1  2(2  1 )G( x)  32G 2 ( x)], x  R
where i  [-1,1], i =1,2 are the transmutation parameters and obey the condition
 2  1  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
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(3)
(4)
(5)
(6)
Journal of Xidian University
https://doi.org/10.37896/jxu16.6/023
ISSN No:1001-2400
F ( x) = 21G( x)  3(2  1 )[G( x)]2  2(1  22  3 )[G( x)]3  (1  1  2  23 )[G( x)]4
with corresponding pdf
(7)
f ( x) = g ( x)[21  6(2  1 )G( x)  6(1  22  3 )G 2 ( x)  4(1  1  2  23 )G3 ( x)], x  R
(8)
where i  [0,1], i =1,2.and 3  [-1,1] are the transmutation parameters and
obey the condition  2  2  3  1. 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)
F ( x) = e w[21  3(2  1 )e w  2(1  22  3 )e2 w  (1  1  2  23 )e3w ]
where x  R,  ,  > 0, i  [0,1], i =1,2 and 3  [-1,1] are the
transmutation parameters and obey the condition  2  2  3  1.
1
It is very important note observe that at value 1 = 2 = 3 = ,the quartic rank
2
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
1
f ( x) = we  w [21  6(2  1 )e  w  6(1  22  3 )e 2 w  4(1  1  2  23 )e 3w ]

(10)
where
(
x
)
w = e  , x  R,  ,  > 0
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.
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Journal of Xidian University
https://doi.org/10.37896/jxu16.6/023
ISSN No:1001-2400
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
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ISSN No:1001-2400
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
R( x) = 1  F ( x). For the quartic rank transmuted Gumbel (QTG) distribution, the
reliability function is given as,
R( X ) = 1  e w[21  3(2  1 )e w  2(1  22  3 )e2 w  (1  1  2  23 )e3w ]
(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,
f ( x)
h( x ) =
,
R( x)
then we can find the hazard rate function of QTG distribution by (11) and (10):
1 w
we [21  6(2  1 )e  w  6(1  22  3 )e 2 w  4(1  1  2  23 )e 3w ]
h( x ) = 
1  e  w [21  3(2  1 )e  w  2(1  22  3 )e 2 w  (1  1  2  23 )e 3w ]
(12)
The cumulative hazard function is defined by
H ( x) = lnR ( x),
so the cumulative hazard function of the QTG distribution is


H ( x) = ln 1  e w[21  3(2  1 )e w  2(1  22  3 )e2 w  (1  1  2  23 )e3w ] .
The reverse hazard function is
f ( x)
r ( x) =
F(X )
Using (14), we can write the reverse hazard function of QTG distribution as
w[21  6(2  1 )e  w  6(1  22  3 )e 2 w  4(1  1  2  23 )e 3w ]
r ( x) =
 [21  3(2  1 )e  w  2(1  22  3 )e 2 w  (1  1  2  23 )e 3w ]
(13)
(14)
(15)
The Odd function of a distribution with cdf F (x) is defined as
F ( x)
O( x) =
1  F ( x)
Then the Odd function of the QTG distribution is given as


O( x) = e w[21  3(2  1 )e  w  2(1  22  3 )e2 w  (1  1  2  23 )e 3w ]
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ISSN No:1001-2400
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
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3.2
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ISSN No:1001-2400
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  1  3  2 . Then the quantile function of X, is given by
xq =    log( log B(q, 1 , 2 , 3 ))
(17)
Proof: To compute the quantile function of the quartic rank transmuted Gumbel
probability distribution, we substitute x by xq and F (x) by q in (9) to get the
equation
q = e e
(
xq  

)
)
[21  3(2  1 )e e
(
xq  

)
)
 2(1  22  3 )e 2e
(18)
Then, we solve the equation (18) for xq . So, let y = e
( e
(
(
xq  

xq  

)
)
 (1  1  2  23 )e 3e
(
xq  

)
)
]
)
)
. Thus, (18) becomes
q = 21 y  3(2  1 ) y  2(1  22  3 ) y  (1  1  2  23 ) y 4
2
3
and hence,
(1  1  2  23 ) y 4  2(1  22  3 ) y 3  3(2  1 ) y 2  21 y  q = 0
(19)
Let a = (1  1  2  23 ), b = 2(1  22  3 ) , c = 3(2  1 ) , d = 21 and e=
-q then the equation (19) becomes
ay 4  by 3  cy 2  dy  e = 0.
Then, equation(20) from [6]
l
 2  4( ) 2  4(    2 z )
b
6
y=

4a
2
where  = z 
m=
2
1
3
l
c 3b
,»» l =  2 ,»» z = (rei ) 
6
a 8a
3
1
(rei ) 3
(20)
m
,»»»»  = 4 and,

b3
d

2
16a
a
Now, let the function B(q, 1 , 2 , 3 ) be defined by
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ISSN No:1001-2400
l
 2  4( ) 2  4(    2 z )
b
6
B(q, 1 , 2 , 3 ) =  
4a
2
Hence,
y = e( e
(
xq  

)
)
= B(q, 1 , 2 , 3 )
(21)
Take natural Logarithm to both sides to get
e
Then, we have the equation
(
xq  

)
= log B(q, 1 , 2 , 3 )
xq =    log( log B(q, 1 , 2 , 3 ))
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
r
r
k
k
E ( x r ) = E ( x r ) = (1) k   k  r k [21 k ( )  6(2  1 ) k (2 ( ))


k =0
k 
 6(1  22  3 )
k

k
(3 ( ))  4(1  1  2  23 )
k

k
(4 ( ))] |  = 1
(22)
Proof: The rth moment of the positive random variable X with probability
density function is given by

E ( x r ) =  x r f ( x)dx
(23)
0
Substituting from (10) in to (23),

1
0

E( xr ) =  xr
we  w [21  6(2  1 )e  w  6(1  22  3 )e 2 w  4(1  1  2  23 )e 3w ]dx
(24)
where w = e
(
x

)
, Then dw =
e
(

x

)
dx , »» x =   lnw
With substitution in (24) using Gamma integration;
r
r
k
k
E ( x r ) = (1) k   k  r k [21 k ( )  6(2  1 ) k (2 ( ))


k =0
k 
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 6(1  22  3 )
k

k
(3 ( ))  4(1  1  2  23 )
ISSN No:1001-2400
k

k
(4 ( ))] |  = 1
Moment Generating Function
Theorem 3.3 The moment generating function M x (t ) of a random variable X
QTGD (  ,  ) is given by

r
tr r
k
k
M x (t ) =  (1) k   k  r k [21 k ( )  6(2  1 ) k (2 ( ))


r = 0 r! k = 0
k 
 6(1  22  3 )
k

k
(3 ( ))  4(1  1  2  23 )
k
k
(4 ( ))] |  = 1
(25)
Proof:The moment generating function of the positive random variable X
with probability density function is given by

M x (t ) = E (etx ) =  etx f ( x)dx
0
tx
Using series expansion of e ,
n
tr  r
tr
M x (t ) =   x f ( x)dx =  E ( x r )
0
r = 0 r!
r = 0 r!
n
Then
r
tr r
k
k
(1) k   k  r k [21 k ( )  6(2  1 ) k (2 ( ))



r = 0 r! k = 0
k 

M x (t ) = 
 6(1  22  3 )
3.4
k

k
(3 ( ))  4(1  1  2  23 )
k
k
(4 ( ))] |  = 1
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,  x (t ) ,is

(it ) r
r = 0 r!
x (t ) = 
r
k
 
k =0
 6(1  22  3 )
k

Where i = 1 and t  
3.5
r
(1)  k 
k
k
 r k [21
k

k
( )  6(2  1 )
(3 ( ))  4(1  1  2  23 )
k
k
k
k
(2 ( ))
(4 ( ))] |  = 1
Cumulant Generating Function
The cumulant generating function is defined by
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K x (t ) = loge M x (t )
Cumulant generating function of quartic rank transmuted Gumbel distribution is
given by

r
tr r
k
k
K x (t ) = log e  (1) k   k  r k [21 k ( )  6(2  1 ) k (2 ( ))


r = 0 r! k = 0
k 
 6(1  22  3 )
4
k

k
(3 ( ))  4(1  1  2  23 )
k
k
(4 ( ))] |  = 1
(27)
ENTROPY
4.1
Rényi Entropy
If X is a non-negative continuous random variable with pdf f (x), then the
Rényi entropy of order  (See Rényi [16]) of X is defined as

1
H  ( x) =
log  [ f ( x)] dx,  > 0, (  1)
0
1 
(28)
Theorem 4.1 The Rényi entropy of a random variable X  QTGD (  ,  ),
with 1  1, 2  0 , 3  0 and 1  2  3 is given by
H  ( x) =
 j k
1
1
log[c( j, k , r ,  )  1 1  j (2  1 ) j k (1  1  2  23 ) r
1 

j =0 k =0 r =0
 (1  22  3 ) k r
( )
]
(  k  j  r )
Proof: Suppose X has the pdf in (10). Then, one can calculate
[ f ( x)] =
1



(29)

(30)
w e w [21  6(2  1 )e  w  6(1  22  3 )e 2 w  4(1  1  2  23 )e 3w ]

By the general binomial expansion, we have
[2  6(
1
2

 1 )ew  6(1  22  3 )e2 w  4(1  1  2  23 )e3w ]

 
=  (21 )  j 6(2  1 )e  w  6(1  22  3 )e 2 w  4(1  1  2  23 )e 3w
j =0 j 
by the Binomial Theorem,


6(
2
 1 )ew  6(1  22  3 )e2 w  4(1  1  2  23 )e3w

j
 j
=   6(2  1 )e  w
k =0 k 
VOLUME 16, ISSUE 6, 2022
 6(  2
j k
1

2

j
j
 3 )e 2 w  4(1  1  2  23 )e 3w
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
k
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by the Binomial Theorem,
6(  2
1
2

 3 )e2 w  4(1  1  2  23 )e3w
k
k 
=   6(1  22  3 )e 2 w
r =0  r 
 4(1    
k r
1
2

k
 23 )e 3w

r
k
k 
=  6k r (1  22  3 ) k r e 2(k r ) w 4r (1  1  2  23 ) r e 3rw
r =0  r 
k
k 
=  6k r 4r (1  1  2  23 ) r (1  22  3 ) k r e (2 k  r ) w
r =0  r 
Substitute from(32) in (31), to get
6(
2

j
 j
=   6(2  1 )e  w
k =0 k 
 1 )ew  6(1  22  3 )e2 w  4(1  1  2  23 )e3w
  kr 6
 
k
j k
 r =0
k r
(32)

j

4r (1  1  2  23 ) r (1  22  3 ) k r e (2 k  r ) w 

j k
 j  k 
=   6 j r (4) r (2  1 ) j k (1  1  2  23 ) r (1  22  3 ) k r e ( k  j  r ) w
k = 0 r = 0  k  r 
Now, substitute from(33) in (30), to get
[2  6(
1
2

 1 )ew  6(1  22  3 )e2 w  4(1  1  2  23 )e3w ]

j k
 
 
 j  k 
=  (21 )  j [  6 j r (4) r (2  1 ) j k (1  1  2  23 ) r
j =0 j 
k = 0 r = 0  k  r 
 (1  22  3 ) k r e( k  j r ) w ]
 j k 
  j  k 
=    2  j 6 j r (4) r 1  j (2  1 ) j k (1  1  2  23 ) r (1  22  3 ) k r e ( k  j  r ) w
j = 0 k = 0 r = 0  j  k  r 
(34)
Now, substitute from(34) in (29), to get
1  w  j k    j  k    j j r r   j

[ f ( x)] =  w e    2 6 (4) 1 (2  1 ) j k

j = 0 k = 0 r = 0  j  k  r 
 (1  1  2  23 ) r (1  22  3 ) k r e( k  j r ) w
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   j  k 
Let c( j, k, r ,  )=    2  j 6 j r (4) r , then
 j  k  r 
1  j k
[ f ( x)] =  c( j, k , r ,  )1  j (2  1 ) j k (1  1  2  23 ) r

j =0 k =0 r =0
 (1  22  3 ) k r e
To find H  (x) , we substitute from (35) in (28)
H  ( x) =
1
1
log [ 
1 


j
k
c( j, k , r,  )
j
1
 (   k  j  r ) w  ( 
( x )

)
(35)
(2  1 ) j k (1  1  2  23 ) r (1  22  3 ) k r
j =0 k =0 r =0
  (   k  j  r ) w  ( 
 e
( x )

)
0
(36)
dx].
We can evaluate the integration

  (  k  j  r ) w  ( 
0
where w = e
(
x

e
)
( x )

)

dx =  (e
(
( x )

)
0
, and   1, then dw =
e
(

x

)  e (  k  j  r ) w dx
)
dx and 0 < w < 
With substitution with this transformation in (37) using Gamma integration,

( )
   w 1e (  k  j  r ) w dw = 
0
(  k  j  r )
After solving the integral, we get the Rényi entropy of the QTGD (  ,  ) by substitute
from (38) in (36)
 j k
1
1
H  ( x) =
log[c( j, k , r ,  )  1 1  j (2  1 ) j k (1  1  2  23 ) r
1 

j =0 k =0 r =0
 (1  22  3 ) k r
4.2
(37)
(38)
( )
]
(  k  j  r )
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 
I H (q) =
1   f ( x) q dx , whereq > 0, andq  1


0
1 q 
Theorem 4.2 The q-entropy of a random variable X  QTGD (  ,  ), with
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1  1, 2  0, 3  0 and 1  2  3 . is given by
I H (q) =
 j k
1
1
[1  c( j, k , r , q) q 1 1q  j (2  1 ) j k (1  1  2  23 ) r
1 q

j =0 k =0 r =0
 (1  22  3 ) k r
( q )
]
(q  k  j  r ) q
Proof. To find I H (q) , we substitute (35) in (40).
4.3
Shannon Entropy
The (Shannons entropy [17]) of a non-negative continuous random variable X
with pdf f (x) is defined as

H ( f ) = E[ log f ( x)] =  f ( x) log( f ( x))dx
0
Below, we are going to use the Expansion of the Logarithm function (Taylor series at 1),
m

m 1 ( x  1)
log ( x) = (1)
, | x |< 1
m
m=0
The Shannon entropy of a random variable X  QTGD (  ,  ), with 1  1, 2 
0, 3  0 and 1  2  3 . is given by
 m n 1 j k
 m 1
1
H ( f ) = (1) n   c( j, k , r , n  1) n1 1n1 j (2  1 ) j k

m =1 n = 0 j = 0 k = 0 r = 0
nm
(n  1)
 (1  1  2  23 ) r (1  22  3 ) k r
(n  1  j  k  r ) n1
Proof: By the Expansion of the Logarithm function (41)

( f ( x)  1) m
log ( f ( x)) = (1) m1
,
m
m =1
and by Binomial Theorem,


 m
1m
= (1) m1 (1) mn   f n ( x)
m  n=0
m =1
n

 m 1
log ( f ( x)) = (1) n 1   f n ( x)
m =1 n = 0
nm
To compute the Shannon’s entropy of X, substitute from (44) in (40)

(40)
(41)
(42)
(43)
m
(44)
 m


 m 1
H ( f ) =   f ( x) log ( f ( x))dx =   f ( x)(1) n1   f n ( x)dx
0
0
m =1 n = 0
nm
H( f ) = 
 
0
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m
 m 1
(1)  n  m f
m =1 n = 0
n
 
n 1
( x)dx
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substituting from (35) in (45), to get
H( f ) = 
n 1 j k
1 n1 j
n  m 1
j k


(

1)


 n  m c( j, k , r , n  1)  n1 1 (2  1 )
m =1 n = 0
  j =0 k =0 r =0
 
0
m
 (1  1  2  23 ) r (1  22  3 ) k r e
( n 1 k  j  r ) w (  ( n 1)
( x )

)
dx
 m n 1 j k
 m 1
1
H ( f ) = (1) n   c( j, k , r , n  1) n1 1n1 j (2  1 ) j k

m =1 n = 0 j = 0 k = 0 r = 0
nm
 ( n 1 k  j  r ) w (  ( n 1)
 (1  1  2  23 ) r (1  22  3 ) k r  e
0
( x )

)
dx
Now, we use (38) to fined the value of the integration, so
 m n 1 j k
 m 1
1
H ( f ) = (1) n   c( j, k , r , n  1) n1 1n1 j (2  1 ) j k

m =1 n = 0 j = 0 k = 0 r = 0
nm
 (1  1  2  23 ) r (1  22  3 ) k r
5
(n  1)
(n  1  j  k  r ) n1
ORDER STATISTICS
Let X 1 , X 2 ,..., X n be a random sample of size k from the QTG distribution with
parameters  > 0 ,  > 0 , 0  1  1 , 1  1  2  1 and 2  1  3  2 From [5],
the pdf of the k th order statistics is obtain by
 n
f X ( x) = k   f ( x)[ F ( x)]k 1[1  F ( x)]nk
(k )
k 
Let X k be the k th order statistic from X  QTGD (  ,  ) with 1  0 , 2  0
and 3  0 . Then, the pdf of the k th order statistic is given by
n
( x) = k   f ( x)[ F ( x)]k 1[1  F ( x)]nk , byBinomialTheorem,
(k )
k 
 nk

n
n  k 
[ F ( x)] j 
= k   f ( x)[ F ( x)]k 1 (1) j 
k 
 j 
 j =0

nk
n
n

k
 

 f ( x)[ F ( x)]k  j 1
= (1) j k  
j =0
 k  j 
fX
nk
 n  n  k  1  w
 we [21  6(2  1 )e  w
( x) = (1) j k  
(k )
j =0
 k  j  
2 w
 6(1  22  3 )e  4(1  1  2  23 )e3w ]
fX
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

 ew[21  3(2  1 )ew  2(1  22  3 )e2 w  (1  1  2  23 )e3w ]
Then, the pdf of first order statistic X (1) of QTG distribution is given as
f X ( x) = n
(1)
1

k  j 1
we  w [21  6(2  1 )e  w  6(1  22  3 )e 2 w  4(1  1  2  23 )e 3w ]


 1  ew[21  3(2  1 )ew  2(1  22  3 )e2 w  (1  1  2  23 )e3w ]
Therefore, the of the largest order statistic X (n ) of QTG distribution is given by
n 1
fX
( n)
( x) = n
1

we  w [21  6(2  1 )e  w  6(1  22  3 )e 2 w  4(1  1  2  23 )e 3w

 ew[21  3(2  1 )ew  2(1  22  3 )e2 w  (1  1  2  23 )e3w
6

n 1
MAXIMUM LIKELIHOOD ESTIMATION (MLE)
Assume X 1 , X 2 ,..., X n be a random sample of size n from QTGD(  ,  ) then
the likelihood function can be written as
n
n
l (  ,  , 1 , 2 , 3 ) =  f ( xi ) = [
i =1
i =1
1

 6(1  22  3 )e
where wi = e
Then
x 
( i
)

wi e
2 wi
 wi
[21  6(2  1 )e
 wi
 4(1  1  2  23 )e
3wi
]]
, i = 1,2,..., n .
l (  ,  , 1 , 2 , 3 ) =
 6(1  22  3 )e
1

n
2 wi
n
wi e
 wi
n
[[2  6(
1
i =1
2
 1 )e
 wi
(47)
i =1
 4(1  1  2  23 )e
3wi
]]
Then, the Log likelihood function of a vector of parameters given as,
n
n
i =1
i =1
log l (  ,  , 1 , 2 , 3 ) = log  f ( xi ) = n log   [
 log[21  6(2  1 )e
 wi
 6(1  22  3 )e
2 wi
xi  

 wi
 4(1  1  2  23 )e
3 wi
]]
Differentiate w.r.t parameters  ,  , 1 , 2 and 3 we have
 log l n 1
= 

 
VOLUME 16, ISSUE 6, 2022
n
w
(48)
i
i =1
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
1

n
https://doi.org/10.37896/jxu16.6/023
3(2  1 ) wi e i  6(1  22  3 ) wi e
   3(
i =1
1
2
w
 1 )e
 wi
2 wi
 3(1  22  3 )e
 2 wi
ISSN No:1001-2400
 6(1  1  2  23 )e
3 wi
 2(1  1  2  23 )e
3 wi
x 
 log l  n n xi  
=

 wi i 2
2

 i =1 

n

i =1
(49)
( xi   )[3(2  1 ) wi e i  6(1  22  3 ) wi e
w
 2 (1  3(2  1 )e
 wi
 3(1  22  3 )e
2 wi
 2 wi
w
 6(1  1  2  23 )e
 2(1  1  2  23 )e
2 w
3 wi
3 wi
]
)
3 w
n
 log l
1  3e i  3e i  2e i
=
 wi
2 w
3 w
1
 3(1  22  3 )e i  2(1  1  2  23 )e i
i =1 1  3(2  1 )e
w
2 w
3 w
n
 log l
3e i  6e i  2e i
=
 wi
2 w
3 w
2
 3(1  22  3 )e i  2(1  1  2  23 )e i
i =1 1  3(2  1 )e
2 w
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.
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,
VOLUME 16, ISSUE 6, 2022
(51)
3 w
n
 log l
3e i  4e i
=
 wi
2 w
3 w
3
 3(1  22  3 )e i  2(1  1  2  23 )e i
i =1 1  3(2  1 )e
7
(50)
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(52)
Journal of Xidian University
https://doi.org/10.37896/jxu16.6/023
ISSN No:1001-2400
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 (-2 L ( ) ): 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 (-2 L ( ) ) , 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
Data
Set1
set2
VOLUME 16, ISSUE 6, 2022
1: Descriptive Statistics of Data Set 1, 2
mean
1.59
29.01
Median
1.506825
25
Skewness
0.89993
3.0962617
kurtosis
3.92376
13.3250012
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ISSN No:1001-2400
Table 2: MLE’s of the parameters and respective SE’s for various distributions for Data
Set 1
Distribution
Parameter


1
2
3


1
2





QTGD
CTGD
TGD
GD
Table
ModeI
QTGD
CTGD
TGD
GD
3:
Estimate
1.02455
0.258847
0.241234
0.208044
-0.885227
1.11008
0.282105
-0.305918
-1.69408
1.5819
0.488214
0.999999
1.33313
0.376428
SE
0.00359295
0.000343268
0.0425602
0.287866
0.574946
0.00244777
0.000685827
0.103321
0.75491
-0.00316457
0.000263098
-0.177738
0.00255327
0.00104844
Goodness-of fit statistics using the selection criteria values for Data Set 1
-2 L ( )
34.935
41.6642
50.701
61.0358
AIC
44.935
49.6642
56.701
65.058
(a)
Fig.5 (a):Fitted pdf for Data Set 1
VOLUME 16, ISSUE 6, 2022
BIC
55.651
58.237
63.130
69.322
HQIC
42.042
47.350
54.965
63.879
AICc
45.988
50.354
57.108
65.258
(b)
(b) Empirical cdf and theoretical cdf for Data Set 1
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CAIC
60.651
62.237
66.130
71.322
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ISSN No:1001-2400
Table 4: MLE’s of the parameters and respective SE’s for various distributions for Data
Set 2
Distribution
Parameter

QTGD
CTGD
TGD
GD
Table
ModeI
QTGD
CTGD
TGD
GD

1
2
3


1
2





Estimate
28.573
10.1077
1
1
-0.38496
27.5443
9.2936
1
-0.444086
26.6412
9.15995
0.677728
23.7077
8.02217
SE
-0.198022
0.412069
-0.0374996
0.0048907
0.060622
-0.522687
0.140624
-0.0929894
-0.0911491
0.374464
0.235455
0.00668499
0.260879
0.148993
5: Goodness-of fit statistics using the selection criteria values for Data Set 3
-2 L ( )
AIC
BIC
HQIC
AICc
CAIC
1898.234
1908.234
1926.114
1906.826
1908.467
1931.114
1939.172
1947.172
1961.476
1946.046
1974.326
1965.476
1951.112
1957.112
1967.839
1956.267
1957.204
1970.839
1976.482
1980.482
1978.634
1979.919
1980.528
1989.634
(a)
Fig.6 (a):Fitted pdf for Data Set 2
VOLUME 16, ISSUE 6, 2022
(b)
(b) Empirical cdf and theoretical cdf for Data Set 2.
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Journal of Xidian University
8.
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ISSN No:1001-2400
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.
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