Journal of Xidian University
https://doi.org/10.37896/jxu16.6/023
ISSN No:1001-2400
Quartic Rank Transmuted Gumbel Distribution
DoaaELhertaniy
PhD student at the Department of Mathematics, University of the Holy
Quran and Taseel of Science, Sudan
Abstract- In this paper, a Quartic Rank Transmuted Gumbel distribution (QTGD) to an ex- tended 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 statis- tics. 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
g(X; µ, σ) = we−w,
(1)
σ
where
x−µ
w = e−( σ ), σ > 0, µ ∈ R.
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and
ISSN No:1001-2400
G(X; µ, σ) = 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 distri- bution (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 + λ)G(x) − λ[G(x)]2,
|λ| ≤ 1
(3)
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 + λ) − 2λG(x)],
|λ| ≤ 1
(4)
where G(x) and g(x) are the cdf and pdf respectively of the base distribution. It is very important observe
that at λ = 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 λ1 and λ2 if its cumulative distribution function (cdf) is given by
F (x) = (1 + λ1)G(x) + (λ2 − λ1)[G(x)]2 − λ2[G(x)]3
(5)
with corresponding pdf
f (x) = g(x)[1 + λ1 + 2(λ2 − λ1)G(x) − 3λ2G2(x)], x ∈ R
(6)
where λi [-1,1],
i=1,2 are the transmutation parameters and obey the condition 2 λ1 +−λ2 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 λ1 , λ2 and λ3 if its cumulative distribution function (cdf) is given by
F (x) = 2λ1G(x) + 3(λ2 − λ1)[G(x)]2 + 2(λ1 − 2λ2 + λ3)[G(x)]3 + (1 − λ1 + λ2 − 2λ3)[G(x)]4
(7)
with corresponding pdf
f (x) = g(x)[2λ1 + 6(λ2 − λ1)G(x) + 6(λ1 − 2λ2 + λ3)G2(x) + 4(1 − λ1 + λ2 − 2λ3)G3(x)], x ∈ R (8)
where λi ∈ [0,1], i=1,2.and λ3 ∈ [-1,1] are the transmutation parameters and obey the condition
−2 ≤λ2 + λ3 ≤ 1. This paper is organized as follows, the Quartic transmuted Gumbel distribution is
defined in Section2. In Section3statistical properties have been discussed, like shapes of the density and
hazard rate functions, quantile function, moments and moment-generating function, Character- istic
Function , and cumulant generating function. Entropy in Section4, for the Quartic transmuted Gumbel
distribution along with the distribution of order statistics in Section5. Section6provides 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 section7. Finally, in Section8, some
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ISSN No:1001-2400
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[2λ1 + 3(λ2 − λ1)e−w + 2(λ1 − 2λ2 + λ3)e−2w + (1 − λ1 + λ2 − 2λ3)e−3w]
(9)
where x ∈ R, µ, σ > 0, λi ∈ [0,1], i=1,2 and λ3 ∈ [-1,1] are the transmutation parameters and obey
the condition −2 ≤ λ2 + λ3 ≤ 1.
It is very important note observe that at value λ1 = λ2 = λ3 = 1 ,the quartic rank transmuted Gumbel
2
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[2λ + 6(λ2 − λ1)e−w + 6(λ1 − 2λ2 + λ3)e−2w + 4(1 − λ1 + λ2 − 2λ3)e−3w]
(10)
1
σ
where
x−µ
w = e−( σ ), x ∈ R, µ, σ > 0
Figure1, show the PDF and CDF of the quartic rank transmuted Gumbel distribution for different values of
parameters λ1, λ2and λ3 where µ=3 and σ=2.
(a)
(b)
Figure 1: The PDF and CDF of CTGD for different value of λ1 , λ2 and λ3 where µ = 3 and σ = 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 R(x)= 1
− F (x). For the quartic
rank transmuted Gumbel (QTG) distribution, the reliability function is given as,
R(X) = 1 − e−w[2λ1 + 3(λ2 − λ1)e−w + 2(λ1 − 2λ2 + λ3)e−2w + (1 − λ1 + λ2 − 2λ3)e−3w]
VOLUME 16, ISSUE 6, 2022
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(11)
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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
h(x) =
σ
we−w[2λ1 + 6(λ2 − λ1)e−w + 6(λ1 − 2λ2 + λ3)e−2w + 4(1 − λ1 + λ2 − 2λ3)e−3w]
(12)
1 − e−w[2λ1 + 3(λ2 − λ1)e−w + 2(λ1 − 2λ2 + λ3)e−2w + (1 − λ1 + λ2 − 2λ3)e−3w]
The cumulative hazard function is defined by
H(x) = −lnR(x),
so the cumulative hazard function of the QTG distribution is
.
Σ
H(x) = −ln 1 − e−w [2λ1 + 3(λ2 − λ1 )e−w + 2(λ1 − 2λ2 + λ3 )e−2w + (1 − λ1 + λ2 − 2λ3 )e−3w ] .
The reverse hazard function is
r(x) = f (x)
F (X)
Using (14), we can write the reverse hazard function of QTG distribution as
r(x) =
w[2λ1 + 6(λ2 − λ1)e−w + 6(λ1 − 2λ2 + λ3)e−2w + 4(1 − λ1 + λ2 − 2λ3)e−3w]
σ[2λ1 + 3(λ2 − λ1)e−w + 2(λ1 − 2λ2 + λ3)e−2w + (1 − λ1 + λ2 − 2λ3)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
,−
Σ function of the QTG distribution is given as
Σ−
O(x) = e−w [2λ1 + 3(λ2 − λ1 )e−w + 2(λ1 − 2λ2 + λ3 )e−2w + (1 − λ1 + λ2 − 2λ )e−3w ]
−1
1
1
3
(16)
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 distri- bution
with parameters σ > 0, µ > 0, 1 λ1 −
1 and
Then≤the quantile function of X, is given by
≤ λ2 ≤1 λ3 λ2. −
xq = µ − σ log(−
(17)
≤log B(q, λ1, λ2, λ3))
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
x −µ
−( q
)
σ
[2λ1 +3(λ2 −λ1 )e−e
)
x −µ
−( q
)
σ
)
+2(λ1 −2λ2 +λ3)e−2e
Then, we solve the equation (18) for xq. So, let y = e(−e
x −µ
−( q σ )
x −µ
−( q
)
σ
)
+(1−λ1 +λ2 −2λ3)e−3e
x −µ
−( q
))
]
(18)
σ
)
. Thus, (18) becomes
q = 2λ1y + 3(λ2 − λ1)y2 + 2(λ1 − 2λ2 + λ3)y3 + (1 − λ1 + λ2 − 2λ3)y4
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(a)
(b)
Figure 2: Reliability and Hazard function of the QTGD for different value of λ1,λ2 and λ3 whereµ =
3 and σ =2
and hence,
(1 − λ1 + λ2 − 2λ3)y4 + 2(λ1 − 2λ2 + λ3)y3 + 3(λ2 − λ1)y2 + 2λ1y − q = 0
(19)
Let a = (1 − λ1 + λ2 − 2λ3), b = 2(λ1 − 2λ2 + λ3), c =3(λ2 − λ1) , d = 2λ1 and e= -q then the
equation (19) becomes
ay4 + by3 + cy2 + dy + e = 0.
Then, equation(20) from [6]
.
l
24(ξ) −
−2ξ +
b
6 4( + β +
(20)
y=−
+
4a 2z)
2
.
2
1
where ξ = z − l ,6 l = c −a 3b ,8az2 = (reiθ ) −
3
3
(re iθ
)3
1
−
4
β= m
ξ
,
3
and, m = − b 16a
+ d2
a
Now, let the function B(q,λ1,λ2,λ3) be defined by
.
Hence,
B(q, λ1, λ2, λ3) = −
4a
y = e(−e
b
+
−2ξ +
4(ξ)2 − 4( l + β + 2z)
6
2
x −µ )
−( qσ
)
= B(q, λ , λ , λ )
1
2
(21)
3
Take natural Logarithm to both sides to get
−( xq−µ )
−e
σ
= log B(q, λ1, λ2, λ3)
Then, we have the equation
xq = µ − σ log(− log B(q, λ1, λ2, λ3))
The first quartile, median and third quartile can be obtained by setting q = 0.25, 0.50 and 0.75 in
(17) respectively.
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3.3. Moments and Moment-generating function
Moments function
Theorem 3.2. Let X ∼ QTGD(µ,σ). Then the rth moment of X is given by
. Σ
∂k
∂k
Σ
r
k r
E(xr) = E(xr ) =
(−1)
σkµr−k[2λ1 k Γ(α) + 6(λ2 − λ1) k (2−αΓ(α))
α
α
k
k=0
+6(λ1 − 2λ 2 + λ )3∂k (3−αΓ(α)) + 4(1 − λ 1 + λ 2− 2λ )3∂k (4−αΓ(α))]|α = 1
(22)
α
α
k
k
Proof: The rth moment of the positive random variable
X with probability density function is given by
∞
E(xr) = ∫
xrf (x)dx
(23)
0
Substituting from (10) in to (23),
∫ ∞ r 1 −w
E(xr ) =
x we [2λ1 + 6(λ2 −λ1)e−w + 6(λ1 − 2λ2 + λ3)e−2w + 4(1 −λ 1
σ
+λ2 − 2λ3)e−3w]dx (24)
0
where w = e−(
x−µ
−e
σ )
−(x−µ
dx, x = µ − σlnw
σ
With substitution in (24) using Gamma integration;
. Σ
∂k
∂k
r
Σ
k r
E(xr) =
(−1)
σkµr−k[2λ1 k Γ(α) + 6(λ2 − λ1) k (2−αΓ(α))
α
α
k
σ
)
, Then dw =
k=0
+6(λ1 − 2λ 2 + λ )3∂k (3−αΓ(α)) + 4(1 − λ
α
k
Moment Generating Function
1
∂k
+ λ 2− 2λ )3 α(4−αΓ(α))]|α = 1
k
Theorem 3.3. The moment generating function Mx(t) of a random variable X QTGD (µ,σ) is given by
Σ r r
. Σ
∂k
∂k −α
Mx(t) = ∞ Σ (−1)k r σkµr−k[2λ1 Γ(α) + 6(λ2 − λ1)
(2 Γ(α))
r!
k
αk
αk
r=0 t k=0
+6(λ1 − 2λ 2 + λ )3∂k (3−αΓ(α)) + 4(1 − λ 1 + λ 2− 2λ )3∂k (4−αΓ(α))]|α = 1
α
α
k
k
Proof:The moment generating function of the positive random variable X with probability density
function is given by
∫ ∞ tx
Mx (t) = E(etx) =
e f (x)dx
(25)
0
tx
Using series expansion of e ,
Mx(t) =
Then
Σ
n
r=0
tr
r!
∫
∞
0
n
Σ
tr
x f (x)dx =
E(xr)
r!
r=0
r
Σ tΣ
∂k
∂k −α
. Σ
(2 Γ(α))
Γ(α) + 6(λ2 − λ1)
∞ r r
r
α
α
k r−k
r k=0 (−1)k
k
σ
µ
[2λ
1 k
r=0
k
!
+6(λ1 − 2λ 2 + λ )3∂kα(3−αΓ(α)) + 4(1 − λ 1 + λ 2− 2λ )3∂kα(4−αΓ(α))]|α = 1
Mx(t) =
k
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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 (µ,σ), then characteristic
function,ϕx(t),is
∞
. Σ
∂k
∂k
Σr r
k r
k r−k
ϕx(t) = Σ (it)
(−1)
(2−αΓ(α))
Γ(α)
+
6(λ
−
λ
)
σ
µ
[2λ
2
1
1 k
k
r!
α
α
k
k=0
r=0
+6(λ 1 − 2λ 2+ λ )3
√
Where i = −1 and t ∈ ℜ
∂k
α
k
(3−αΓ(α)) + 4(1 − λ +
1 λ −2 2λ )
∂k
3
α
k
(4−αΓ(α))]|α = 1
(26)
3.5. Cumulant Generating Function
The cumulant generating function is defined by
Kx(t) = loge Mx(t)
Cumulant generating function of quartic rank transmuted Gumbel distribution is given by
∞
. Σ
∂k
∂k
Σ
r r
k r
k r−k
Kx(t) = loge Σ t
(−1)
(2−αΓ(α))
Γ(α)
+
6(λ
−
λ
)
σ
µ
[2λ
2
1 k
1 k
r!
α
α
k
k=0
r=0
+6(λ1 − 2λ 2 + λ )3∂k (3−αΓ(α)) + 4(1 − λ
α
k
1
∂k
+ λ 2− 2λ )3 α(4−αΓ(α))]|α = 1
k
(27)
4. ENTROPY
4.1.Rényi Entropy
If X is a non-negative continuous random variable with pdf f (x), then the Rényi entropy of order δ
(See Rényi [16]) of X is defined as
∫∞
1
Hδ(x) =
log [f (x)]δdx, ∀δ > 0, (δ ̸= 1)
(28)
1−
0
δ
Theorem 4.1. The Rényi entropy of a random variable X ∼ QTGD (µ,σ), with λ1 =
̸
1, λ2 ̸= 0
,λ3 ̸= 0 and λ1 ̸= λ2 ̸= λ3 is given by
∞
Hδ (x) =
1
1−
δ
j
k
ΣΣΣ
log[
j=0 k=0 r=0
−1
j−k
c(j, k, r, δ) δ−1 λδ−j
(λ 2 − λ 1) (1 − λ 1 + λ 2− 2λ )3
1
σ
Γ(δ)
×(λ1 − 2λ 2 + λ 3)k−r (δ + k + j + r)δ
]
Proof: Suppose X has the pdf in (10). Then, one can calculate
1
Σ
[f (x)]δ = wδe−δw [2λ
+ 6(λ2 − λ1)e−w + 6(λ1 − 2λ2 + λ3)e−2w + 4(1 − λ1
1
δ
σ
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r
+ λ2 − 2λ 3)e−3w ]
Σδ
(29)
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By the general binomial expansion, we have
Σ
−3w
+
λ
]
2 − 2λ )e
Σ
−2w
δ
+ 4(1 − λ1
[2λ1 + 6(λ2 − λ1)e−w + 6(λ1 − 2λ2 + λ3)e
3
∞. Σ
Σ
Σ
Σ
δ
−w
−3w j
2λ 2 + λ 3)e−2w + 4(1 − λ +1λ
=
(2λ )1δ−j 6(λ 2 − λ 1)e + 6(λ −
2 − 2λ 3)e
1
j
j=0
by the Binomial Theorem,
Σ
−w
−2w
+ 4(1 − λ1
6(λ2 − λ1)e + 6(λ1 − 2λ2 + λ3)e
+ λ2 − 2λ )e−3w
3
j
Σ
. Σ
Σ
j Σ 6(λ2 − λ1)e−w k j− Σ
=
− 2λ2 + λ3)e−2w + 4(1 − λ1
k
6(λ
1
k=0
by the Binomial Theorem,
Σ
−2w
+ 4(1 − λ1
6(λ1 − 2λ2 + λ3)e
+ λ2
Σj
Σ
−3w
+ λ2 − 2λ3)e
− 2λ )e−3w
(30)
k
(31)
Σk
3
k
Σ
. Σ
Σ
Σ
Σ r
k Σ 6(λ1 − 2λ2 + λ3)e−2w k−r
−3w
r
4(1
−
λ
+
λ
−
2λ
)e
1
2
3
r=0
k . Σ
Σ k k−r
=
6 (λ1 − 2λ2 + λ3)k−re−2(k−r)w4r(1 − λ1 + λ2 − 2λ3)re−3rw
r
r=0
k . Σ
Σ k k−r r
6 4 (1 − λ1 + λ2 − 2λ3)r(λ1 − 2λ2 + λ3)k−re−(2k+r)w
=
r
r=0
Substitute from(32) in (31), to get
Σ
Σ
−w
−2w
+ 4(1 − λ1 + λ2 − 2λ )e−3w j
6(λ2 − λ1)e + 6(λ1 − 2λ2 + λ3)e
3
=
j
Σ k . Σ
Σ
. Σ
Σ
j Σ 6(λ2 − λ1)e−w k j− Σ k 6k−r4r(1 − λ1 + λ2 − 2λ3) r(λ1 − 2λ2
=
k
r
r=0
k=0
(32)
Σ
+ λ3)k−re−(2k+r)w
j
Σ
Σk . j Σ.k Σ j−r r
6
(4) (λ2 − λ1 )j−k (1 − λ1 + λ2 − 2λ3 )r (λ1 − 2λ2 + λ3 )k−r e−(k+j+r)w
=
k r
(33)
k=0 r=0
Now, substitute from(33) in (30), to get
Σ
Σ
−w
−2w
+ 4(1 − λ1 + λ2 − 2λ 3)e−3w ] δ
[2λ1 + 6(λ2 − λ1)e + 6(λ1 − 2λ2 + λ3)e
j Σ
k
Σ
. Σ. Σ j−r r
Σ .δΣ (2λ1 )δ−j [
(4) (λ2 − λ1 )j−k (1 − λ1 + λ2 − 2λ3 )r
j k 6
= ∞ j
k r
j=0
k=0 r=0
×(λ1 − 2λ2 + λ3)k−re−(k+j+r)w]
∞
Σ
Σj Σk
=
. Σ. Σ. Σ δ−j j−r r δ−j
δ j k 2 6 (4) λ (λ2 −λ1 )j−k (1−λ1 +λ2 −2λ3 )r (λ1 −2λ2 +λ3 )k−r e−(k+j+r)w
1
j k
r
j=0 k=0 r=0
(34)
Now, substitute from(34) in (29), to get
j Σ
∞ Σ
k . Σ. Σ. Σ
Σ
δ j k
[f (x)]δ = 1 wδe−δw
2δ−j 6j−r (4)r λ1δ−j (λ2 − λ1 )j−k
δ
σ
j k
r
j=0 k=0 r=0
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×(1 − λ1 + λ2 − 2λ3 )r (λ1 − 2λ2 + λ3 )k−r e−(k+j+r)w
Let c( j, k, r ,δ)=
.δΣ. j Σ.kΣ δ−j j−r r
2 6 (4) , then
j k r
∞
δ
[f (x)] =
j
k
1 ΣΣΣ
σδ
j−k
δ−
j
(1 − λ1
2 − λ1 )
1c(j,
+ λ2 − 2λ3) r
j=0 k=0 r=0
k,r,δ)λ
(λ
(x−µ)
×(λ1 − 2λ2 + λ3)k− e−(δ+k+j+r)w+(−δ ) σ
r
To find Hδ(x), we substitute from (35) in (28)
∞
H δ (x) =
1
1−
δ
log
j
k
Σ1 ΣΣΣ
σ
δ
(35)
c(j, k, r, δ)λδ−j
(λ 2 − λ 1)j−k (1 − λ 1+ λ
1
2
− 2λ3) r(λ
1
− 2λ 2 + λ 3)k−r
j=0 k=0 r=0
∫
∞
×
e−(δ+k+j+r)w+(−δ
(x−µ)
)
Σ
σ
dx
.
(36)
0
We can evaluate the integration
∫ ∞
∫
σ
−(δ+k+j+r)w+(−δ (x−µ))
0 e
dx =
−(x−µ
where w = e
)
σ
, and δ ≥ 1, then dw =
∞
(e(−0
(x−µ)
x−µ
−e−( σ )
σ
σ
) × e−(δ+k+j+r)w dx
) δ
(37)
dx and 0 < w < ∞
With substitution with this transformation in (37) using Gamma integration,
∫∞
(38)
Γ(δ)
−σ
w δ−1 e−(δ+k+j+r)w dw = −σ
0
(δ + k + j + r)δ
After solving the integral, we get∞thej Rényi
entropy of the QTGD (µ,σ) by substitute from (38) in (36)
k
Σ
Σ
Σ
Σ
1
−1
Hδ (x) =
log
c(j, k, r, δ) δ−1λδ−j
(λ 2 − λ 1)j−k (1 − λ 1 + λ 2− 2λ )3 r
1
σ
1−
j=0 k=0 r=0
δ
Σ
Γ(δ)
×(λ1 − 2λ 2 + λ 3)k−r
δ
(δ + k + j + r)
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
Σ
Σ
∫∞
1
q
IH(q) =
1−
f (x) dx , where q > 0, and q ̸= 1
(39)
1−
0
q
0,λ3 ̸= 0
Theorem 4.2. The q-entropy of a random variable X ∼ QTGD (µ,σ), with λ1 ̸= 1, λ2
and λ1 ̸= λ2 ̸= λ3. is given by
IH (q) =
1
1−
q
Σ
∞
1−
j
k
ΣΣΣ
c(j, k, r, q)
j=0 k=0 r=0
−1 q−j
λ 1 (λ 2 − λ 1)j−k (1 − λ 1+ λ
q−1
σ
×(λ1 − 2λ 2 + λ 3)k−r
Γ(q)
(q + k + j + r)q
2
− 2λ 3) r
Σ
Proof. To find IH(q), we substitute (35) in (40).
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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[− logf (x)] = −
f (x)log(f (x))dx
(40)
0 function (Taylor series at 1),
Below, we are going to use the Expansion of the Logarithm
∞
m
Σ
log(x) =
(−1)
− 1)
, |x| < 1
m
m−1 (x
m=0
(41)
The Shannon entropy of a random variable X ∼ QTGD (µ,σ), with λ1 ≠
1, λ2 ̸= 0,λ3
λ1 ̸= λ2 ̸= λ3. is given by
Σ∞ m n+1 j
k
. Σ
1 n+1−j
ΣΣΣΣ
H(f ) =
(−1)n m 1 c(j, k, r, n + 1)
λ
(λ2 − λ1)j−k
1
n m
σn+1
0 and
(42)
m=1 n=0 j=0 k=0 r=0
r 2λ
× (1 − λ 1+ λ −2 2λ ) (λ
−
3
1
Γ(n +1)
+ λ 3)k−r (n + 1 + j + k + r)n+1
2
(43)
Proof: By the Expansion of the Logarithm function (41)
∞
Σ
log(f (x)) =
(−1)
m−1 (f (x) − 1)
m
m=1
m
,
and by Binomial Theorem,
.
∞
Σ
=
(−1)
m−1 1
m
m=1
log(f (x)) =
m
Σ
(−1)
m−n
n=0
Σ∞ Σ
m
Σ
. Σ
m n
f (x)
n
. Σ
m 1 f n (x)
n m
(−1)n+1
m=1 n=0
(44)
To compute the Shannon’s entropy of X, substitute from (44) in (40)
∫∞
∫∞
. Σ
Σ
m
∞ Σ
H(f ) = −
f (x)log(f (x))dx = −
f (x)
(−1)n+1 m 1 fn(x)dx
n m
0
0
m=1 n=0
. Σ
∫ ∞Σ
∞ Σ
m
m 1
−
H(f )=
substituting from (35) in (45), to get
∫
∞ Σ
m
∞Σ
H(f )=
0
m=1 n=0
0
m=1 n=0
( 1)n
n+1
(x)dx
(45)
. Σ Σ
n+1 jΣ Σ
k
1
n+1−j
(−1)n m 1
c(j, k, r, n + 1)
(λ2 − λ1)j−k
1λ
n+1
n m
σ
j=0 k=0 r=0
× (1 − λ1 + λ2 − 2λ3) (λ1r − 2λ2 + λ3)
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n mf
229
k−
r
e−(n+1+k+j+r)w+(−(n+1)
(x−µ)
)
dx
σ
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H(f ) =
Σ∞
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k
. Σ
n+1 jΣ Σ
m Σ
1
Σ
(−1)n m 1 c(j, k, r, n + 1)
λn+1−j (λ2 − λ1)j−k
n m
σn+1 1
m=1 n=0 j=0 k=0 r=0
∫
r
× (1 − λ1 + λ 2− 2λ )3 (λ
1
− 2λ 2 + λ 3)
k−r
∞
0
e−(n+1+k+j+r)w+(−(n+1)
(x−µ)
)σ
dx
Now, we use (38) to fined the value of the integration, so
H(f ) =
Σ∞
. Σ
m n+1 j
k
1
Σ Σ Σ Σ(−1)n m 1 c(j, k, r, n + 1)
λn+1−j (λ2 − λ1)j−k
1
n+1
n m
σ
m=1 n=0 j=0 k=0 r=0
× (1 − λ 1+λ
2
Γ(n +1)
− 2λ3) r(λ 1 − 2λ 2 + λ 3)k−r (n + 1 + j + k + r)n+1
5. ORDER STATISTICS
Let X1, X2, ..., Xn be a random sample of size k from the QTG distribution with parameters µ > 0, σ >
0, 0 λ1 1≤, λ1 ≤
1 λ2 λ1−
and ≤
λ2 1 ≤λ3 λ2 From [5],
− the
≤ pdf of the k th order statistics is obtain by
≤
. Σ
fX(k)
n
(x) = k k f (x)[F (x)]k−1 [1 − F (x)]n−k
(46)
Let Xk be the k th order statistic from X
pdf of the k th order statistic is given by
∼ QTGD (µ,σ) withλ1
0,λ2 ̸= 0 and λ3 ̸= 0. Then, the
. Σ
fX(k) (x) = k n f (x)[F (x)]k−1 [1 − F (x)]n−k , by Binomial Theorem,
k
Σ
Σ n−k
.
Σ
. Σ
Σ
n
f (x)[F (x)]k−1
(−1)j n − k [F (x)]j
=k
j
k
j=0
. Σ.
Σ
(−1)j k n n − k f (x)[F (x)]k+j−1
k
j
j=0
. Σ.
Σ
n−k
fX (x) = Σ(−1)j k n n − k 1 we−w[2λ1 + 6(λ2 − λ1)e−w
(k)
k
j
σ
j=0
=
n−k
Σ
+ 6(λ1 − 2λ2 + λ3)e−2w + 4(1 − λ1 + λ2 − 2λ3)e−3w]
Σ
Σk+j−1
× e−w [2λ1 + 3(λ2 − λ1 )e−w + 2(λ1 − 2λ2 + λ3 )e−2w + (1 − λ1 + λ2
− 2λ 3)e−3w ]
Then, the pdf of first order statistic X(1) of QTG distribution is given as
1 −w
(x) =
[2λ1 + 6(λ2 − λ1)e−w + 6(λ1 − 2λ2 + λ3)e−2w + 4(1 − λ1 + λ2 − 2λ3)e−3w]
n σwe
fX(1)
Σ
.
ΣΣ
−w
−2w
+ (1 − λ1 + λ2 − 2λ )e−3w] n−1
× 1 − e−w [2λ1 + 3(λ2 − λ1)e + 2(λ1 − 2λ2 + λ3)e
3
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Therefore, the of the largest order statistic X(n) of QTG distribution is given by
1 −w
(x) =
[2λ1 + 6(λ2 − λ1)e−w + 6(λ1 − 2λ2 + λ3)e−2w + 4(1 − λ1 + λ2 − 2λ3)e−3w
n σwe
fX(n)
Σ
Σ
−w
−2w
+ (1 − λ1 + λ2 − 2λ 3)e−3w n−1
× e−w [2λ1 + 3(λ2 − λ1)e + 2(λ1 − 2λ2 + λ3)e
6. MAXIMUM LIKELIHOOD ESTIMATION (MLE)
Assume X1, X2, ..., Xn be a random sample of size n from QTGD(µ,σ) then the likelihood function can
be written as
Y
Y
n
n Σ
l(µ, σ, λ1, λ2, λ3) =
f (xi) =
1 wie−wi [2λ1 + 6(λ2 − λ1)e−wi
i=1
i=1
σ
Σ
−2wi
−3wi
+6(λ1 − 2λ2 + λ3)e
+ 4(1 − λ1 + λ2 − 2λ3)e
]
xi−µ
)
σ
where wi = e−(
Then
,
i = 1, 2, ..., n.
Σ
l(µ, σ, λ1, λ2, λ3) = 1n
σ
n
Σ
[2λ1 + 6(λ2 − λ1)e−wi
wie−wi n
Y
i=1
−2wi
+ 6(λ1 − 2λ2 + λ3)e
−3wi
+ 4(1 − λ1 + λ2 − 2λ3)e
Log likelihood function of a vector of parameters given as,
Y
Σn Σ
n
log l(µ, σ, λ , λ , λ ) = log
f (x ) = −n log σ +
1
2
i
3
i=1
−wi
+ log[2λ1 + 6(λ2 − λ1)e
(47)
i=1
−
Σ
] Then, the
xi − µ
− wi
σ
i=1
−2wi
+ 6(λ1 − 2λ2 + λ3)e
+ 4(1 − λ1 + λ2 − 2λ3)e
−3wi
Σ
] Differentiate w.r.t
parameters µ , σ , λ1, λ2 and λ3 wehave
n
∂ log l n 1 Σ
(48)
= −
w i
∂µ
σ σ i=1
−2wi
n
w
−3wi
− 6(1 − λ +1λ
1 Σ 3(λ 2 − λ 1)w ie i − 6(λ 1 − 2λ 2 + λ 3)w ei
2 − 2λ3 )e
+
σ i=1 λ1 + 3(λ2 − λ1)e−wi + 3(λ1 − 2λ2 + λ3)e−2wi + 2(1 − λ1 + λ2 − 2λ3)e−3wi
n
∂ log l −n Σ xi − µ
xi − µ
(49)
=
+
−w i
2
2
σ
σ
∂
σ
i=1
σ
−3wi
n
wi
Σ
+ λ 3)w ei −2wi − 6(1 − λ 1 + λ 2− 2λ )e
]
(x −
i µ)[3(λ −
2 λ )w
1 ei − 6(λ 1 − 2λ 2
3
+
σ2(λ1 + 3(λ2 − λ1)e−wi + 3(λ1 − 2λ2 + λ3)e−2wi + 2(1 − λ1 + λ2 − 2λ3)e−3wi )
i=1
n
∂ log l Σ
=
∂λ1
i=1
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1 − 3e−wi + 3e−2wi − 2e−3wi
λ1 + 3(λ2 − λ1)e−wi + 3(λ1 − 2λ2 + λ3)e−2wi + 2(1 − λ1 + λ2 − 2λ3)e−3wi
231
(50)
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∂ log l = Σ
∂λ2
i=1
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λ1 + 3(λ2 − λ1
)e−wi
ISSN No:1001-2400
3e−wi − 6e−2wi + 2e−3wi
+ 3(λ1 − 2λ2 + λ3)e−2wi + 2(1 − λ1 + λ2 − 2λ3)e−3wi
(51)
n
Σ
(52)
∂ log l =
3e−2wi − 4e−3wi
i=1
∂λ3
λ1 + 3(λ2 − λ1)e−wi + 3(λ1 − 2λ2 + λ3)e−2wi + 2(1 − λ1 + λ2 − 2λ3)e−3wi
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 simulta- neously.
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 (µ
, σ , λ1, λ2,and λ3) 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 max- imum of
monthly wind speed (mph) in Falconara Marittima, Ancona, Italy for the months Jan- uary 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 Table1. 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 Tables2and4for
selected models using the MLE method.
In Tables3and5, 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(θ) ): where is L(θ) 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(θ)) , 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
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ISSN No:1001-2400
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
Set1
set2
mean
1.59
29.01
Median Skewness
1.506825 0.89993
25
3.0962617
kurtosis
3.92376
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
µ
1.02455
0.00359295
σ
λ1
0.258847
0.000343268
0.241234
0.0425602
λ2
0.208044
0.287866
λ3
-0.885227
0.574946
µ
1.11008
0.00244777
σ
λ1
0.282105
0.000685827
-0.305918
0.103321
λ2
-1.69408
0.75491
µ
1.5819
-0.00316457
σ
0.488214
0.000263098
λ
0.999999
-0.177738
µ
1.33313
0.00255327
σ
0.376428
0.00104844
CTGD
TGD
GD
Table 3: Goodness-of fit statistics using the selection criteria values for Data Set 1
ModeI
QTGD
CTGD
TGD
GD
VOLUME 16, ISSUE 6, 2022
-2L(θ)
AIC
34.935 44.935
41.6642 49.6642
50.701 56.701
61.0358 65.058
BIC
55.651
58.237
63.130
69.322
233
HQIC
42.042
47.350
54.965
63.879
AICc
45.988
50.354
57.108
65.258
CAIC
60.651
62.237
66.130
71.322
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Distribution
Parameter
Estimate
(a)
(b)SE
Fig.5 (a):Fitted pdf for Data Set 1
(b) Empirical cdf and theoretical cdf for Data Set 1.
QTGD
28.573
-0.198022
µ
Table 4: MLE’s of the parameters and respective SE’s for various distributions for Data Set 2
10.1077
0.412069
σ
λ1
1
-0.0374996
CTGD
TGD
GD
λ2
1
0.0048907
λ3
-0.38496
0.060622
µ
27.5443
-0.522687
σ
λ1
9.2936
0.140624
1
-0.0929894
λ2
-0.444086
-0.0911491
µ
26.6412
0.374464
σ
9.15995
0.235455
λ
0.677728
0.00668499
µ
23.7077
0.260879
σ
8.02217
0.148993
Table 5: Goodness-of fit statistics using the selection criteria values for Data
ModeI
QTGD
CTGD
TGD
GD
-2L(θ)
1898.234
1939.172
1951.112
1976.482
AIC
1908.234
1947.172
1957.112
1980.482
BIC
1926.114
1961.476
1967.839
1978.634
HQIC
1906.826
1946.046
1956.267
1979.919
AICc
1908.467
1974.326
1957.204
1980.528
Set 3
CAIC
1931.114
1965.476
1970.839
1989.634
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, relia-
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(a)
Fig.6 (a):Fitted pdf for Data Set 2
ISSN No:1001-2400
(b)
(b) Empirical cdf and theoretical cdf for Data Set 2.
bility 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 , trans- muted Gumbel and
cubic transmuted Gumbel distributions for the applied data sets.
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