Multivariate Modified Dugum Distribution and Its Applications

Abstract

The modified Dagum distribution is a highly versatile statistical model, and it is included in several important parametric families of distributions, with applications, such as economics and public health. In this paper, we introduce a multivariate version of the modified Dagum distribution and deduce some of its sub-models to address specific analytical needs. We use two different approaches to derive the joint probability density function for the proposed distribution. Also, we derive the joint cumulative distribution function through the traditional method and the Clayton copula methods. In addition, we explore and discuss some statistical properties, including the multivariate dependence. Further, we use the maximum likelihood method to estimate the unknown parameters and the associated confidence interval. Finally, we apply the proposed model to analyze some real data sets, including a protein consumption data set and a warranty policy data set, for demonstrative purposes. The marginals of the proposed model fit the data sets quite well, and the results demonstrate the model’s effectiveness in modeling the proposed data.

Full text

Academic Editor: Stelios Psarakis
Received: 21 April 2025
Revised: 12 May 2025
Accepted: 13 May 2025
Published: 15 May 2025
Citation: Alghufily, N.; Sultan, K.S.;
Radwan, H.M.M. Multivariate
Modified Dugum Distribution and Its
Applications. Mathematics 2025,13,
1620. https://doi.org/10.3390/
math13101620
Copyright: © 2025 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license
(https://creativecommons.org/
licenses/by/4.0/).
Article
Multivariate Modified Dugum Distribution and Its Applications
Naelah Alghufily 1,* , Khalaf S. Sultan 2and Hossam M. M. Radwan 3
1Department of Mathematical Sciences, College of Science, Princess Nourah Bint Abdulrahman University,
P.O. Box 84428, Riyadh 11671, Saudi Arabia
2Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City, Cairo 11884, Egypt;
ksultan@azhar.edu.eg
3Mathematics Department, Faculty of Science, Minia University, Minia 61519, Egypt;
hmmradwan86@mu.edu.eg
*Correspondence: nmalghufily@pnu.edu.sa
Abstract: The modified Dagum distribution is a highly versatile statistical model, and it
is included in several important parametric families of distributions, with applications,
such as economics and public health. In this paper, we introduce a multivariate version of
the modified Dagum distribution and deduce some of its sub-models to address specific
analytical needs. We use two different approaches to derive the joint probability density
function for the proposed distribution. Also, we derive the joint cumulative distribution
function through the traditional method and the Clayton copula methods. In addition,
we explore and discuss some statistical properties, including the multivariate dependence.
Further, we use the maximum likelihood method to estimate the unknown parameters and
the associated confidence interval. Finally, we apply the proposed model to analyze some
real data sets, including a protein consumption data set and a warranty policy data set, for
demonstrative purposes. The marginals of the proposed model fit the data sets quite well,
and the results demonstrate the model’s effectiveness in modeling the proposed data.
Keywords: multivariate modified Dagum distribution; Clayton copula; multivariate
dependence properties; maximum likelihood estimation; protein consumption and
warranty policies data sets
MSC: 62H10; 62F10; 60E05
1. Introduction
The IWD is a versatile tool that can be easily applied to model various processes across
different fields. These include reliability, ecology, medicine, biological studies, public health,
and economics. The distribution has been extensively explored in the literature, where its
properties and wide-ranging applications have been discussed. Notable contributions can
be found in the works of [1–4].
The CDF and PDF of the IWD with parameters (λ,α)is given, respectively, by
G(x;λ,α) = e−1
λxα,x>0, (1)
and
g(x;λ,α) = α
λ1
xα+1
e−1
λxα,x>0, (2)
where λ>0 is the scale parameter and α>0 is the shape parameter.
Mathematics 2025,13, 1620 https://doi.org/10.3390/math13101620

Mathematics 2025,13, 1620 2 of 27
The GIWD was first introduced by [
5
] and has proven to be an effective model for
a wide range of applications. It is particularly useful in fields such as medicine and bio-
logical studies, where it can accurately represent various processes. Its flexibility makes
it a valuable tool in these disciplines, providing a reliable framework for modeling com-
plex phenomena.
The CDF and PDF of the GIWD with parameters (λ,α,ξ)are given, respectively, by
G(x;λ,α,ξ) = e−ξ
λxα,x>0, (3)
and
g(x;λ,α,ξ) = α ξ
λ1
xα+1
e−ξ
λxα,x>0, (4)
where λ>0 and ξ>0 are the scale parameters and α>0 is the shape parameter.
The authors of [
6
] successfully defined the three-parameter modified Weibull distribu-
tion by extending the traditional Weibull distribution. This extension involved the inclusion
of an additional term,
e−γx
. Similarly, the authors of [
7
] developed a new extended Weibull
distribution by modifying the Weibull distribution through the addition of the term
e−γ
x
.
Following a comparable approach, it is possible to derive the MGIWD by adding the term
e−γx
. Then, the CDF and PDF of the MGIWD with parameters
(λ
,
α
,
ξ
,
γ)
can be given,
respectively, as
G(x;λ,α,ξ,γ) = e−ξ
λxαe−γx,x>0, (5)
and
g(x;λ,α,ξ,γ) = ξ
λ1
xα+1
(γx+α)e−γxe−ξ
λxαe−γx,x>0, (6)
where
λ>
0 and
ξ>
0 are the scale parameters and
α>
0 and
γ≥
0 are the
shape parameters.
The DD, which was first proposed by [
8
], has found several important applications
across a variety of fields. In finance, actuarial sciences, and economics, it plays a key role
in modeling the distribution of personal income, where it is highly valued for its ability
to represent income size distribution. The DD, which is the most well-known model in
economics, is essentially a BIIID with an added scale parameter and a sub-model of the
GB2, as highlighted by [9].
The CDF and PDF of DD with parameters (b,a,ϕ)are given, respectively, as
W(x;b,a,ϕ) = 1
1+bx−aϕ,x>0, (7)
and
w(x;b,a,ϕ) = bϕa x−a−1
1+bx−aϕ+1,x>0, (8)
where b>0 is a scale parameter and a>0 and ϕ>0 are the shape parameters.
A generalization for DD has been proposed by a number of authors in order to produce
more adaptable models. The authors of [
10
] presented a new model named gamma Dagum

Mathematics 2025,13, 1620 3 of 27
distribution, and the Weibull Dagum distribution was proposed by [
11
]. By explicitly
interpreting the Dagum family’s parameters in terms of the income median, inequality,
and poverty measures, ref. [12] proposed novel formulations of the Dagum family, which
are widely appreciated for modeling the income distribution. In [
13
], the exponentiated
generalized exponential Dagum distribution was proposed and investigated. One of the
generalizations that is the subject of this study is the MDD, which was studied by [
14
].
It is also considered a generalization of both MBIIID [
15
] and the modified Fr
´
echet dis-
tribution [
16
]. The CDF and PDF of MDD with parameters
(b
,
a
,
ϕ
,
c)
can be written,
respectively, as
W(x;b,a,ϕ,c) = 1
1+bx−ae−cxϕ,x>0, (9)
and
w(x;b,a,ϕ,c) = bϕe−cx x−a−1(cx +a)
1+bx−ae−c x ϕ+1,x>0, (10)
where b>0 is a scale parameter and a>0, c>0, and ϕ>0 are the shape parameters.
A range of CDF forms that can be useful for fitting data was outlined by [
17
]. These
forms provide different approaches for modeling and analyzing data in various contexts.
Among the various CDFs discussed, one notable example is the BIIID, which is often
employed due to its versatility and ability to fit different types of data effectively. The BIIID
has a wide range of applications in statistical modeling. It is particularly useful in various
fields where statistical analysis is essential, including reliability, forestry, and engineering;
see, for example, [
18
–
20
]. The CDF and PDF of BIIID with parameters
(a
,
ϕ)
, which are a
special case of the CDF and PDF for DD when
b=
1 in Equations (7) and (8), can be written,
respectively, as
W(x;a,ϕ) = 1
1+ ( 1
x)aϕ,x>0, (11)
and
w(x;a,ϕ) = aϕ
xa+11+ ( 1
x)aϕ+1,x>0. (12)
There are various techniques available for introducing multivariate distributions. One
of the earliest approaches was proposed by [
21
], who introduced the multivariate Pareto
distribution. This particular distribution is characterized by having Pareto marginals,
which are individual Pareto distributions for each variable in the multivariate context.
Another technique for obtaining multivariate distributions is introduced by [
22
]. In his
paper, a method to derive the multivariate Burr Type-XII distribution is presented. This
distribution is notable for deriving Burr Type-XII marginals, meaning that each individual
marginal follows a Burr Type-XII distribution. Additionally, ref. [
23
] proposed a new
class of multivariate distributions conducted by compounding the likelihood function of
a specific distribution with a gamma distribution. Also, by applying a suitable transfor-
mation to Takahashi’s multivariate Burr distribution [
22
], ref. [
24
] introduce a new type of
distribution known as the MVGED. Ref. [
25
] introduces two distinct techniques for creating

Mathematics 2025,13, 1620 4 of 27
multivariate versions of the GB2 distribution. The first technique emphasizes stochastic
dependency through gamma random variables, while the second relies on generalizing the
distribution of the order statistics.
Copulas are powerful multivariate statistical modeling tools that depict the depen-
dence structure between random variables independently of their marginal distributions.
According to Sklar’s Theorem (see [
26
]), each multivariate distribution can be decomposed
into its marginal distributions and an associated copula function. Also, the concept of a
copula is derived from the separation of the joint CDF into two parts: one that explains
the dependent structure and the other that describes the marginal behavior. Among the
various families of copulas, the Clayton copula is widely used for modeling asymmetric
lower tail dependence. Let the k-variate random vector
Ω=Ω1
,
. . .
,
Ωk
; then, the k-variate
Clayton copula can be given as
Cθ(ω1, . . . , ωk) = 1
(ω−1/θ
1+. . . +ω−1/θ
k−(k−1))θ, (13)
where
ω1
,
. . .
,
ωk∈[
0, 1
]
are uniformly distributed marginal variables, and
θ>
0 is the
copula parameter controlling the strength of dependence. As
θ→
0, the Clayton copula
approaches independence.
This paper uses two methods suggested in [
21
,
22
] to obtain the joint PDF for the
MVMDD and its sub-models. Furthermore, it is possible to demonstrate that the joint CDF
for MVMDD can be achieved for the first approach by using the conventional approach,
and the second approach can be obtained by using the Clayton copula. Several properties
are discussed for MVMDD. The maximum likelihood estimation cannot be obtained in
closed form, so a numerical method can be used. The effectiveness of the suggested model
and the approach in a practical setting has been demonstrated using two real data sets.
The structure of the paper is organized as follows. Section 2presents the derivation of
some necessary prerequisites. In Section 3, the MVMDD and its sub-models are introduced.
Section 4explores various statistical properties of these distributions. Section 5investigates
the multivariate dependence properties for MVMDD. The maximum likelihood estimation
method is discussed in Section 6. Two applications within the public health field and
warranty policies are illustrated in Section 7. Finally, Section 8provides concluding remarks
and future work.
2. Prerequisites
In this section, the PDF of MDD can be obtained by another approach using the
methodology proposed by [22] as follows:
Suppose
λ>
0 is a random variable with an inverse gamma distribution with PDF;
u(λ;θ,β)is provided by
u(λ;θ,β) = βθ
Γ(θ)1
λθ+1
e−β
λ,λ>0, (14)
using the methodology suggested by [
22
], the PDF of the MDD can be given by MGIWD
(6) with inverse gamma distribution (14) as compounded as

Mathematics 2025,13, 1620 5 of 27
f(x;θ,β,α,γ) = Z+∞
−∞u(λ;θ,β)g(x;λ,α,ξ,γ)dλ
=Z∞
0
βθ
Γ(θ)1
λθ+1
e−β
λξ
λ1
xα+1
(γx+α)e−γxe−ξ
λxαe−γxdλ
= (α+γx)ξe−γxβθ
Γ(θ)1
xα+1Z∞
01
λθ+2
e−1
λ(β+ξx−αe−γx)dλ
= (α+γx)ξe−γxβθ
Γ(θ)1
xα+1Γ(1+θ)
β+ξx−αe−γxθ+1
= (α+γx)ξ θ e−γxβθ1
xα+11
β+ξx−αe−γxθ+1. (15)
The PDF for MDD, which is given by (10), can be obtained from (15) by putting
ξ=b
,
θ=ϕ,γ=c,α=a, and β=1.
3. Multivariate Modified Dagum Distribution
In this section, the joint PDF and the joint CDF of the MVMDD can be derived through
two distinct approaches. Each approach offers a unique technique to calculate the joint PDF
and the joint CDF, allowing for flexibility depending on the chosen technique. Utilizing both
methods can lead to a deeper and more comprehensive understanding of the distribution’s
behavior. Additionally, the joint survival function can also be derived.
3.1. Approach (1): The Joint PDF
Consider a multivariate modified generalized inverse Weibull distribution (MVMGIWD),
where each marginal follows a generalized inverse Weibull distribution given by Equation (6).
The joint density function of the MVMGIWD can be expressed as
g(x;α;ξ;γ,λ) =
k
∏
i=1 ξi(γixi+αi)1
λ1
xiαi+1
e−γxe
−ξie−γixi
λxαi
i!, (16)
where
x= (x1
,
. . .
,
xk)
,
α= (α1
,
. . .
,
αk)
,
ξ= (ξ1
,
. . .
,
ξk)
, and
γ= (γ1
,
. . .
,
γk)
. Following
the technique described in [
22
], we integrate the joint PDF of MVMGIWD with regard to an
inverse gamma mixing distribution. This yields the MVMDD, which captures dependencies
between variables. This leads us to the following lemma, which formally presents the joint
PDF of the MVMDD.
Lemma 1. Assume that
α1
,...,
αn≥
1,
θ≥
0,
ξ1
,...,
ξn>
0and
γ1
,...,
γn>
0. Consider
x1
,
. . .
,
xn
to have a multivariate MGIWD with joint PDF given by (16). Then, the joint PDF for
MVMDD can be derived as
f(x;α;ξ;γ;θ) = Γ(k+θ)
Γ(θ)
e−∑k
i=1γixi∏k
i=1ξi(γixi+αi)x−αi−1
i
1+∑k
i=1ξix−αi
ie−γixi!k+θ,xi>0, i=0, 1, . . . , k. (17)
Proof.
Similar to the method outlined in [
22
], the MVMDD can be derived by compounding
the MGIWD, as defined in Equation (16), with an inverse gamma distribution, as defined

Mathematics 2025,13, 1620 6 of 27
in Equation (15). This approach integrates out the scale parameter to introduce flexibility
and dependence across variables. The resulting joint PDF of the MVMDD is given as
f(x;α;ξ;γ;θ) = Z∞
0
k
∏
i=1 ξi(γixi+αi)1
λ1
xiαi+1
e−γixie
−ξie−γixi
λxαi
i!
×βθ
Γ(θ)1
λθ+1
e−β
λdλ
=βθ
Γ(θ)
k
∏
i=1 ξi(γixi+αi)1
xiαi+1
e−γixi!
×Z∞
0e
−1
λ β+∑k
i=1
ξie−γixi
xαi
i!1
λk+θ+1
dλ
=
k
∏
i=1 ξi(γixi+αi)x−αi−1
ie−γixi!βθ
Γ(θ)Γ(k+θ)
β+∑k
i=1ξix−αi
ie−γixik+θ.
By setting
β=
1, (17) is easily obtained, which is the joint PDF of MVMDD. Also, some
important sub-models can be obtained from Equation (17) as shown in Table 1. Further,
the MVMBIIID can be considered as a multivariate of the MBIIID given in [15].
Table 1. The sub-models from the joint PDF of MVMDD given by (17).
Distribution Parameters Joint PDF
MVDD [25]γ1=0, . . . , γk=0f(x;α;ξ;θ) = Γ(k+θ)
Γ(θ)
∏k
i=1ξiαix−αi−1
i
1+∑k
i=1ξix−αi
i!k+θ
MVMBIIID [New] ξ1=1, . . . , ξk=1f(x;α;γ;θ) = Γ(k+θ)
Γ(θ)
e−∑k
i=1γixi∏k
i=1(γixi+αi)x−αi−1
i
1+∑k
i=1x−αi
ie−γixi!k+θ
MVBIIID [27]γ1=0, . . . , γk=0f(x;α;θ) = Γ(k+θ)
Γ(θ)
∏k
i=1αix−αi−1
i
1+∑k
i=1x−αi
i!k+θ
ξ1=1, . . . , ξk=1
MVLD [27,28]α1=0, . . . , αk=0f(x;γ;θ) = Γ(k+θ)
Γ(θ)
e−∑k
i=1γixi∏k
i=1γi
1+∑k
i=1e−γixi!k+θ
ξ1=1, . . . , ξk=1
3.2. Approach (2): The Joint PDF
The joint PDF of MVMDD can be obtained by another method. This method was
given by [21,24], who used this methodology to obtain the joint PDF of MVGED. Ref. [21]
proposed the following joint PDF:
f(v1, . . . , vk) = θ(θ+1). . . (θ+n−1)
(1+v1+. . . +vk)n+θ, (18)
where
V=V1
,
. . .
,
Vk
is the k-variate random vector,
v1>
0,
. . .
,
v2>
0 and
θ>
0. Then,
upon using the appropriate random vector with the joint PDF (18), the marginals, the joint

Mathematics 2025,13, 1620 7 of 27
CDF, and the joint survival function can be obtained in explicit form. The following lemma
will serve as a basis for deriving the form of the joint PDF for MVMDD based on the
technique of [21].
Lemma 2. For
x1>
0,
. . .
,
xk>
0, the joint PDF for MVMDD given by Equation (17) can
be obtained.
Proof.
Let the k-variate random vector
V=V1
,
. . .
,
Vk
with the joint PDF (18). Consider
the k-variate random vector
Vi=ξix−αi
ie−γixi
; then, the random vector
x
has the joint
PDF (17), where θ(θ+1). . . (θ+n−1) = Γ(k+θ)
Γ(θ).
3.3. The Joint CDF
In this subsection, the traditional technique and the Clayton copula are applied to
derive the joint CDF of MVMDD.
Lemma 3. For
x1>
0, . . . ,
xk>
0, the joint CDF for MVMDD can be given by the traditional
technique as
F(x;α;ξ;γ;θ) = 1
1+∑k
i=1ξix−αi
ie−γixi!θ,xi>0, i=0, 1, . . . , k. (19)
Proof.
Equation (19) can be obtained directly by integrating the Equation (17) with respect
to x1, . . . , xkfrom 0 to xi,i=1, . . . , kas follows.
F(x;α;ξ;γ;θ) = Zxk
0. . . Zx1
0f(x1, . . . , xn;α1, . . . , αn;φ1, . . . , φn;θ)dx1. . . dxk
=Γ(k+θ)
Γ(θ)Zxk
0. . . "Zx1
0e−∑k
i=2γixi
k
∏
i=2
ξi(γixi+αi)x−αi−1
i
×e−γ1x1ξ1(γ1x1+α1)x−α1−1
1
1+∑k
i=2ξix−αi
ie−γixi+ξ1x−α1
1e−γ1x1!k+θdx1#. . . dxk
=Γ(k+θ−1)
Γ(θ)Zxk
0. . . "Zx2
0e−∑k
i=3γixi
k
∏
i=3
ξi(γixi+αi)x−αi−1
i
×e−γ2x2ξ2(γ2x2+α2)x−α2−1
2
1+∑k
i=3ξix−αi
ie−γixi+ξ2x−α2
2e−γ2x2+ξ1x−α1
1e−γ1x1!k+θ−1dx2#. . . dxk
=Γ(k+θ−2)
Γ(θ)Zxk
0. . . "Zx3
0e−∑k
i=4γixi
k
∏
i=4
ξi(γixi+αi)x−αi−1
i
×e−γ3x3ξ3(γ3x3+α3)x−α3−1
3
1+∑k
i=4ξix−αi
ie−γixi+ξ3x−α3
3e−γ3x3+∑2
j=1ξjx−αj
je−γjxj!k+θ−2dx3#. . . dxk
and so on until (19) is obtained.
In the following remark, the Clayton copula is applied to derive the joint CDF of MVMDD.

Mathematics 2025,13, 1620 8 of 27
Remark 1. For
x1>
0, . . . ,
xk>
0, the joint CDF for MVMDD given by Equation (19) can be
obtained by the Clayton copula (see, for example, [
24
,
26
]). The joint CDF for MVMDD can be
obtained using the Clayton copula (13) by considering
F(x1, . . . , xk) = Cθ(F(x1), . . . , F(xk)),
and
F(xi)=(
1
+ξix−αi
ie−γixi)−θ
,
i=
1,
. . .
,
k
are the marginals CDF of MVMDD; then, the
random vector X has the joint CDF for MVMDD is given by (19).
Figures 1–3display a graphical analysis of the bivariate MDD, illustrating its joint
PDF and joint CDF under varying parameter settings to obtain a clear vision of the behav-
ior of the bivariate MDD. These figures demonstrate how parameter changes affect the
distribution’s shape and tail behavior. As the parameters increase, particularly
γ1
and
γ2
,
with fixed values of parameters
α1
,
α2
,
ξ1
, and
ξ2
and a large value of
θ
, Figure 1illustrates
that the distribution exhibits a sharper and narrower PDF peak. In contrast, with fixed
values of parameters
α1
,
α2
,
ξ1
and
ξ2
and a large value of
θ
, lower parameter values for
γ1
and
γ2
produce a flatter and wider PDF, as shown in Figure 1. Additionally, with a
small value of
θ
, one can show that the joint PDF of bivariate MDD has bi-modality or
tri-modality shapes and also has a long right tail, as shown in Figure 2. The various shapes
for the joint CDFs of bivariate MDD with various parameters
(γ1
,
γ2
,
θ)
and fixed values
of parameters
α1
,
α2
,
ξ1
and
ξ2
are shown in Figure 3. From Figure 3, we see that larger
values of the parameter
θ
result in faster accumulation in the joint CDF for bivariate MDD,
and smaller values of the parameter
θ
result in slower accumulation in the joint CDF for
bivariate MDD.
Figure 1. The joint PDFs of the bivariate MDD with different values of the parameters
(γ1=0.5
,
1.5, 2.5,
γ2=
0.5, 1.5, 2.5,
θ=
3) and fixed value of parameters the
α1=
1.5,
α2=
2.5,
ξ1=
3 and
ξ2=
2.5.

Mathematics 2025,13, 1620 9 of 27
Figure 2. The joint PDFs of the bivariate MDD with different values of the parameters
(γ1=
0.5,
1.5,2.5,
γ2=
0.5,1.5, 2.5,
θ=
0.5
)
and fixed values of the parameters
α1=
1.5,
α2=
2.5,
ξ1=
3, and
ξ2=2.5.
Figure 3. The joint CDFs of the bivariate MDD with different values of the parameters
(γ1
= 0.5, 2.5,
γ2=0.5, 2.5)and θ=0.5,3 and fixed values of the parameters α1=1.5, α2=2.5, ξ1=3 and ξ2=2.5.

Mathematics 2025,13, 1620 10 of 27
3.4. Joint Survival Function
The traditional joint survival and the copula joint survival functions for MVMDD are
derived in this subsection. These joint survival functions can be defined, respectively, as
S(x;α;ξ;γ;θ) = P X1≥x1∪. . . ∪Xk≥xk!=1−P X1≤x1∪. . . ∪Xk≤xk!, (20)
and
S(x;α;ξ;γ;θ) = ¯
Cθ(S(x1), . . . , S(xk)). (21)
where
S(x1)
,...,
S(xk)
are the marginal survival functions, which can be obtained from the
marginal CDFs; for more details, see [
26
]. The following lemma presents the joint survival
function for MVMDD and a formula for the joint survival functions based on the copula.
Lemma 4. For
x1>
0, ...,
xk>
0and
S(x1)
, ...,
S(xk)
are the marginal survival functions;
then, the joint survival function for MVMDD can be derived as
S(x;α;ξ;γ;θ) = 1−∑
1≤i≤k
1
(1+ξix−αi
ie−γixi)θ
+∑
1≤i<j≤k
1
(1+ξix−αi
ie−γixi+ξjx−αj
je−γjxj)θ
−. . . + (−1)k+11
(1+∑k
i=1φix−αi
ie−γixi)θ
= (1−k) + ∑
1≤i≤k
S(xi) + ∑
1≤i<j≤k
Cθ(1−S(xi), 1 −S(xj))
−. . . + (−1)k+1Cθ(1−S(x1), . . . , 1 −S(xk)). (22)
Proof.
Equation (22) can be obtained directly using the inclusion–exclusion identity [
29
].
4. Properties of the Proposed Distribution
In this section, some properties of MVMDD are discussed. The marginals of PDF and
CDF for MVMDDs can be obtained in closed form and also classified as the PDF and CDF
of MVMDDs. Additionally, the conditional distribution can be obtained for MVMDD, and
the random samples can be generated from MVMDD.
Lemma 5. For any MVMDD, the following properties are satisfied.
(i)
Every marginal CDF obtained from MVMDD is also classified as the CDF of MVMDD.
(ii)
Every marginal PDF obtained from MVMDD is also classified as PDF of MVMDD.
(iii)
Every conditional distribution obtained from MVMDD is also classified as MVMDD and is
given by

Mathematics 2025,13, 1620 11 of 27
f(x1, . . . , xm|xm+1, . . . , xk) =
Γ(k+θ)e−∑m
i=1γixi∏m
i=1αi+γixix−αi−1
i
Γ(k+θ−m)
×
m
∏
i=1
ξi
1+∑k
j=m+1ξjx−αj
je−γjxj
×1
1+∑m
i=1
ξi
1+∑k
j=m+1ξjx−αj
je−γjxjx−αi
ie−γixi!k+θ
which follows the MVMDD with joint PDF
f(x1, . . . , xm;α1, . . . , αm;ξ1
1+∑k
j=m+1ξjx−αj
je−γjxj, . . . , ξm
1+∑k
j=m+1ξjx−αj
je−γjxj;γ1, . . . , γm;k+θ−m).
Proof. To prove the preceding lemma, the following steps are required:
(i)
The proof for Lemma (5) (i) can be obtained directly from (19) as follows:
F(˜x; ˜α;˜
ξ; ˜γ;θ) = lim
x1→∞
1
1+∑k
i=2ξix−αi
ie−γixi+ξ1x−α1
1e−γ1x1!θ
=1
1+∑k
i=2ξix−αi
ie−γixi!θ,
which is the joint CDF for MVMDD
(x2
,
. . .
,
xk)
, and
˜x= (x2
,
x3
,
. . .
,
xn)
,
˜α=
(α2,α3, . . . , αn),˜
ξ= (ξ2,ξ3, . . . , ξn), and ˜γ= (γ2,γ3, . . . , γn).
(ii)
The proof for Lemma (5) (ii) can be obtained directly by integrating (17) from 0 to
∞
with respect to xi,i=1, 2, 3, . . . , kas follows:
fx1(˜x; ˜α;˜
ξ; ˜γ;θ−1) = Z∞
0
Γ(k+θ)
Γ(θ)e−∑k
i=2γixi
k
∏
i=2
ξi(γixi+αi)x−αi−1
i
×e−γ1x1ξ1(γ1x1+α1)x−α1−1
1
1+∑k
i=2ξix−αi
ie−γixi+ξ1x−α1
1e−γ1x1!k+θdx1,
=Γ(k+θ−1)
Γ(θ)
e−∑k
i=2γixi∏k
i=2ξi(γixi+αi)x−αi−1
i
1+∑k
i=2ξix−αi
ie−γixi!k+θ−1,
which is the joint PDF for MVMDD(x2, . . . , xk).
(iii)
The proof for Lemma (5) (iii) can be obtained directly from (17), and the proof of
Lemma (5) (ii) is as follows:
f(x1, . . . , xm|xm+1, . . . , xk) = f(x1, . . . , xk)
f(xm+1, . . . , xk),

Mathematics 2025,13, 1620 12 of 27
where
f(x1
,
. . .
,
xk)
is given by (17) and
f(xm+1
,
. . .
,
xk)
can be directly obtained from
the proof of Lemma (5) (ii) as follows:
f(xm+1, . . . , xk) = Γ(k+θ−m)
Γ(θ)
e−∑k
j=m+1γjxj∏k
j=m+1ξj(γjxj+αj)x−αj−1
j
1+∑k
j=m+1ξjx−αj
je−γjxj!k+θ−m.
The following lemma can be used to generate random samples from MVMDD.
Lemma 6. Let f (x1, . . . , xk)be the joint PDF of MVMDD, which is given by (17).
(i) The marginal PDF of
x1
and the conditional PDFs of
(x2|x1)
,
(x3|x2
,
x1)
,...,
(xk|x1
,
. . .
,
xk−1)
are, respectively, given by
f(x1) = θe−γ1x1ξ1(γ1x1+α1)x−α1−1
1
1+ξ1x−α1
1e−γ1x1!θ+1,
f(x2|x1) =
(θ+1)e−γ2x2ξ2(γ2x2+α2)x−α2−1
2 1+ξ1x−α1
1e−γ1x1!θ+1
1+∑2
i=1ξix−αi
ie−γixi!θ+2,
f(x3|x1,x2) =
(θ+2)e−γ3x3ξ3(γ3x3+α3)x−α3−1
3 1+∑2
i=1ξix−αi
ie−γixi!θ+2
1+∑3
i=1ξix−αi
ie−γixi!θ+3,
.
.
.
f(xk|x1, . . . , xk−1) =
(θ+k−1)e−γkxkξk(γkxk+αk)x−αk−1
k 1+∑k−1
i=1ξix−αi
ie−γixi!θ+k−1
1+∑k
i=1ξix−αi
ie−γixi!θ+k.
(ii)
The corresponding CDFs for the marginal PDF of
x1
and the conditional PDFs of
(x2|x1)
,
(x3|x2,x1),...,(xk|x1, . . . , xk−1)are, respectively, given by

Mathematics 2025,13, 1620 13 of 27
F(x1) = 1
1+ξ1x−α1
1e−γ1x1!θ,
F(x2|x1) = 1+ξ1x−α1
1e−γ1x1!θ+1
1+∑2
i=1ξix−αi
ie−γixi!θ+1,
F(x3|x1,x2) = 1+∑2
i=1ξix−αi
ie−γixi!θ+2
1+∑3
i=1ξix−αi
ie−γixi!θ+2,
.
.
.
F(xk|x1, . . . , xk−1) = 1+∑k−1
i=1ξix−αi
ie−γixi!θ+k−1
1+∑k
i=1ξix−αi
ie−γixi!θ+k−1.
Proof.
From Lemma 5(ii and iii), Lemma 6(i) can be easily proved. Additionally, Lemma 6
(ii) can be proved from Lemma 6(i) by the traditional method.
5. Multivariate Dependence Properties
In this section, we investigate the positively lower orthant dependent (PLOD) and lift
rail decreasing (LTD) properties for MVMDD. First, we present some definitions for PLOD,
and LTD is introduced; for more details, see [24,26].
Definition 1. A multivariate random vector
(X1
,
. . .
,
Xk)
is said to be PLOD iff
(X1
,
. . .
,
Xk)
satisfies the following inequality:
F(x1, . . . , xk)≥
k
∏
i=1
F(xi),
where F(xi),i=1, . . . , k are the marginal CDFs.
Definition 2. Let i =1, . . . , m and j =m+1, . . . , k. Then, (Xj)is said to be LTD in (Xi)if
P(Xj≤xj|Xi≤xi),
is a non-increasing function of xifor all xj.
Lemma 7. Let
f(x1
,
. . .
,
xk)
be the joint PDF of MVMDD given by (17); then, the random vector
(X1, . . . , Xk)is
(i)
PLOD for MVMDD.
(ii)
LTD for MVMDD.
Proof. To prove Lemma 7, the following steps are required:

Mathematics 2025,13, 1620 14 of 27
(i)
To prove (i), we want to prove that
1+
k
∑
i=1
ξix−αi
ie−γixiθk
∏
i=1
1
1+ξix−αi
ie−γixiθ≤1,
which can be written as
1+
k
∑
i=1
ξix−αi
ie−γixik
∏
i=1
1
1+ξix−αi
ie−γixi≤1,
we use mathematical induction to prove this inequality. For
k=
1, the inequality
is true. Consider the inequality is true when
k=l
. Now, we want to prove that
the inequality is true if
k=l+
1. For
k=l+
1, the following inequality should
be satisfied:
1+
l+1
∑
i=1
ξix−αi
ie−γixil+1
∏
i=1
1
1+ξix−αi
ie−γixi≤1. (23)
To prove the case of k=l+1, we can show from the case of k=lthat
1+
l
∑
i=1
ξix−αi
ie−γixil+1
∏
i=1
1
1+ξix−αi
ie−γixi≤1
1+ξl+1x−αl+1
l+1e−γl+1xl+1, (24)
and
ξl+1x−αl+1
l+1e−γl+1xl+1l+1
∏
i=1
1
1+ξix−αi
ie−γixi≤1−1
1+ξl+1x−αl+1
l+1e−γl+1xl+1. (25)
From (24) and (25), one can show that (1) is satisfied.
(ii)
Using Definition 5.2 to prove (ii), let i=1, . . . , mand j=m+1, . . . , k. Then,
P(Xj≤xj|Xi≤xi) = 1+∑m
i=1ξix−αi
ie−γixi!θ
1+∑k
i=1ξix−αi
ie−γixi!θ
=1
1+∑k
i=m+1ξix−αi
ie−γixi
1+∑m
i=1ξix−αi
ie−γixi!θ, (26)
which is a non-increasing function of x1, . . . , xmfor all xm+1, . . . , xk.
6. Maximum Likelihood Estimation
Suppose that
xi1
,
. . .
,
xij
is an independent random sample of size
s
from
MVMDD(Θ)
,
where
Θ= (α1
,
. . .
,
αk
;
ξ1
,
. . .
,
ξk
;
γ1
,
. . .
,
γk
;
θ)
. From Equation (17), the likelihood function
can be obtained as
L(Θ) = "Γ(k+θ)
Γ(θ)
k
∏
i=1
ξi#n∏k
i=1∏n
j=1x−αi−1
ij (αi+γixij)e−γixij
∏n
j=11+∑k
i=1e−γixij ξix−αi
ij k+θ.

Mathematics 2025,13, 1620 15 of 27
Then, the log likelihood function ℓ(Θ)can be given as
ℓ(Θ) = nlog "Γ(k+θ)
Γ(θ)
k
∏
i=1
ξi#+log
k
∏
i=1
n
∏
j=1
x−αi−1
ij +log
k
∏
i=1
n
∏
j=1
(αi+γixij )
+log
k
∏
i=1
n
∏
j=1
e−γixij −log
n
∏
j=11+
k
∑
i=1
e−γixij ξix−αi
ij k+θ. (27)
By taking the first derivative of (27) with respect to all parameters and equating the result to
zero, the MLEs of the unknown parameters can be obtained by solving the
(3k + 1)
normal
equations. The first derivatives of log-likelihood function can be derived for
i=
1, 2,
. . .
,
k
as
∂ℓ
∂θ =
nΓ(θ)Γ(k+θ)ψ(0)(k+θ)
Γ(θ)−ψ(0)(θ)Γ(k+θ)
Γ(θ)
Γ(k+θ)−
n
∑
j=1
log k
∑
i=1
ξix−αi
ij eγi(−xi j)+1!, (28)
∂ℓ
∂αi=
k
∑
s=1
n
∑
j=1
δi,s
γsxjs +αs+
k
∑
s=1
n
∑
j=1
−δi,slogxjs(29)
−(θ+k)
n
∑
j=1
∑k
s=1ξsδi,sx−αs
js −eγs(−xjs )logxjs 
∑k
i=1ξix−αi
ij eγi(−xi j)+1
,
∂ℓ
∂ξi=nk
∑
s=1
δi,s
ξs−(θ+k)
n
∑
j=1
∑k
s=1δi,sx−αs
js eγs(−xjs )
∑k
i=1ξix−αi
ij eγi(−xi j)+1
, (30)
and
∂ℓ
∂γi=
k
∑
s=1
n
∑
j=1
xjsδi,s
γsxjs +αs−
k
∑
s=1
n
∑
j=1
xjsδi,s(31)
−(θ+k)
n
∑
j=1
∑k
s=1ξsδi,sx1−αs
js −eγs(−xjs )
∑k
i=1ξix−αi
ij eγi(−xi j)+1
,
where
δi,s=(1i=s,
0i=s,
is the Kronecker delta function,
ψ(m)(y) = dm+1
dym+1log Γ(y)
, and
(Γ(y)is the traditional gamma function), which are polygamma functions; see [30].
These (3k + 1) nonlinear equations in (28)–(31) cannot be solved analytically, so they
can be solved numerically by the Newton–Raphson method using Mathematica 11; see
Appendix B.
Fisher Information Matrix for MVMDD
Given the difficulty of computing the Fisher information matrix (which is obtained
by taking the expectation of the second derivative of (27)), it would seem appropriate to
use their MLEs to approximate these expected values. Then, as in [
31
], the asymptotic
variance–covariance matrix is provided.

Mathematics 2025,13, 1620 16 of 27












Var(ˆ
θ)Cov(ˆ
θ,ˆ
α1)Cov(ˆ
θ,ˆ
α2). . . Cov(ˆ
θ,ˆ
ξ1)Cov(ˆ
θ,ˆ
ξ2). . . Cov(ˆ
θ,ˆ
γk)
Cov(ˆ
α1,ˆ
θ)Var(ˆ
α1)Cov(ˆ
α1,ˆ
α2). . .
Cov(ˆ
α2,ˆ
θ)Cov(ˆ
α2,ˆ
α3)Var(ˆ
α2). . .
.
.
.
Cov(ˆ
γk,ˆ
θ)Var(ˆ
γk)












=












−ℓθθ (Θ)−ℓθα1(Θ)−ℓθα2(Θ). . . −ℓθξ1(Θ)−ℓθξ2(Θ). . . −ℓθ γk(Θ)
−ℓα1θ(Θ)−ℓα1α1(Θ)−ℓα1α2(Θ). . .
−ℓα2θ(Θ)−ℓα2α1(Θ)−ℓα2α2(Θ). . .
.
.
.
ℓγkθ(Θ)ℓγkγk(Θ)












−1
(ˆ
θ,ˆ
αl,ˆ
ξl,ˆ
γl)
, (32)
where ℓθiθj(Θ) = ∂2ℓ
∂θiθj,i,j=1, . . . , (3k+1)are derived in Appendix A.
Consequently, for the parameters
Θ
, the approximate CIs derived from the asymptotic
variance–covariance matrix can be obtained from
(ˆ
θl±zα
2qVar(ˆ
θl))
,
l=
1,
. . .
,
(
3
k+
1
)
,
where the standard normal distribution’s percentile with right tail probability
α
2
is
zα
2
.
The ACIs could produce a negative lower bound even when the parameters are positive.
In this sense, zero might be used in place of the negative values.
7. Applications
In order to demonstrate the practical applications and effectiveness of the suggested
model and the estimates, we provide an analysis of two real data sets, including protein
consumption data [
32
] and warranty policy data [
33
]. For every data set, the MVMDD
is compared with its sub-models MVDD, MVBIIID, and MVMBIIID and also with the
following well-known multivariate distribution derived in [34].
The joint PDF of MVIWD proposed by [
34
], which is a multivariate distribution based
on the class proposed by [23], is given as
f(x;α;ξ;θ) = Γ(k+θ)
Γ(θ)
∏k
i=1
ξiαi
θx−αi−1
ieξi
θx−αi
i
1−k+∑k
i=1eξi
θx−αi
i!k+θ,xi>0, i=0, 1, . . . , k.
The K-S test and the associated p-value are calculated for the marginals of the MVMDD, as
well as of the marginals of its sub-models MVDD, MVBIIID, and MVMBIIID, and also of
the marginals of MVIWD to determine how closely these distributions fit the data. This
step is not sufficient to show that the MVMDD, MVDD, MVBIIID, and MVIWD fit the
multivariate data or not, but it can be a guide for fitting (see [
24
]). Furthermore, the AIC is
used to compare the candidate multivariate distributions.
7.1. Protein Consumption Data
Consider the data, which are the measurements in twenty-five European countries of
protein consumption in nine food groups. The multivariate data were collected and are
presented in [
32
], p. 535. In this paper, the data obtained from the measurements for four
food groups (red meat, white meat, milk, and starch), summarized in Table 2, are analyzed

Mathematics 2025,13, 1620 17 of 27
to show the performance of the proposed multivariate distribution by comparing them
with some sub-models, and MVIWD.
Table 2. Protein consumption in twenty-five European countries of data from four food groups.
Food Group Measurements
Red meat 10.1, 8.9, 13.5, 7.8, 9.7, 10.6, 8.4, 9.5, 18.0, 10.2, 5.3, 13.9, 9.0, 9.5, 9.4, 6.9, 6.2, 6.2, 7.1,
9.9, 13.1, 17.4, 9.3, 11.4, 4.4
White meat 1.4, 14., 9.3, 6., 11.4, 10.8, 11.6, 4.9, 9.9, 3.0, 12.4, 10.0, 5.1, 13.6, 4.7, 10.2, 3.7, 6.3,
3.4, 7.8, 10.1, 5.7, 4.6, 12.5, 5.0
Milk 8.9, 19.9, 17.5, 8.3, 12.5, 25.0, 11.1, 33.7, 19.5, 17.6, 9.7, 25.8, 13.7, 23.4, 23.3, 19.3, 4.9,
11.1, 8.6, 24.7, 23.8, 20.6, 16.6, 18.8, 9.5
Starch 0.6, 3.6, 5.7, 1.1, 5.0, 4.8, 6.5, 5.1, 4.8, 2.2, 4.0, 6.2, 2.1, 4.2, 4.6, 5.9, 5.9, 3.1,
5.7, 3.7, 2.8, 4.7, 6.4, 5.2, 3.0
Table 3shows the values of K-S distance with the associated p-value for the marginal
distributions and IWD. From Table 3, one can show that the marginal distributions for
MDD, DD, BIIID, and IWD fit all univariate data sets well based on the values of p-value.
So we can say that the MVMDD, MVDD, MVBIIID, and MVIWD are suitable models for the
multivariate data. The Q-Q plots are used to validate marginal distributions for all data sets
of the protein consumption data before estimating the multivariate distribution. Figure 4
shows the Q-Q plots for four data sets of the protein consumption data. From Figure 4, it
is evident that the marginal distribution of MVMDD fits all four data sets of the protein
consumption data well.
Figure 4. Q-Q plot the four food groups in the protein consumption data.

Mathematics 2025,13, 1620 18 of 27
Table 3. K-S test distance with the associated p-value for the protein consumption data.
Uni-Variate Model Data K-S p-Value
Red meat 0.1087 0.9289
MDD White meat 0.1566 0.5719
Milk 0.1228 0.8450
Starch 0.0775 0.9982
Red meat 0.1057 0.9427
DD White meat 0.1551 0.5841
Milk 0.1222 0.8496
Starch 0.0594 0.9999
Red meat 0.1397 0.7133
BIIID White meat 0.1863 0.3507
Milk 0.1584 0.5571
Starch 0.2528 0.0819
Red meat 0.1402 0.7094
IWD White meat 0.1926 0.3118
Milk 0.1579 0.5607
Starch 0.2748 0.0458
The MLEs for the unknown parameters of MVMDD, MVDD, MBIIID, and MVIWD
are obtained when the multivariate protein consumption data are used. Table 4displays the
values of the MLEs for MVMDD, MVDD, MBIIID, and MVIWD, and the associated values
for AIC. Based on the values of the AICs for MVMDD, MVDD MVBIIID, and MVIWD, it is
evident that MVMDD has the lowest value and is considered better than the other models.
Additionally, the plots of the profile log-likelihood function of the estimated parameters for
the protein consumption data are shown in Figure 5.
Table 4. The estimates of the unknown parameters of MVMDD, MVDD, MBIIID, and MVIWD with
the associated values of AIC for the protein consumption data.
Parameters MVMDD MVDD MVBIIID MVIWD
ˆ
θ1.6308 1.1288 12.739 4.1003
ˆ
α11.0964 4.2861 1.3394 2.9045
ˆ
α20.7791 2.7456 1.5348 1.6371
ˆ
α30.5523 3.2721 1.0836 1.9949
ˆ
α40.0043 2.5119 1.9857 1.4102
ˆ
ξ1178.32 11,304.7 1 388.45
ˆ
ξ219.689 167.517 1 15.147
ˆ
ξ351.726 7260.55 1 153.48
ˆ
ξ417.008 24.6011 1 4.2682
ˆ
γ10.3534 0 0 -
ˆ
γ20.2782 0 0 -
ˆ
γ30.1804 0 0 -
ˆ
γ40.8316 0 0 -
AIC 539.64 554.88 633.29 585.77

Mathematics 2025,13, 1620 19 of 27
Figure 5. The log-likelihood function of the estimated parameters of MVMDD for the protein
consumption data.
The calculated variance–covariance matrix of the unknown parameters based on the
protein consumption data is presented in Appendix A. The 95% and 90% ACIs using
the calculated variance–covariance matrix are computed for the unknown parameters of
MVMDD for the protein consumption data and reported in Table 5.

Mathematics 2025,13, 1620 20 of 27
Table 5. The 95% and 90% ACIs of the unknown parameters of MVMDD for the protein consump-
tion data.
Parameters 95%ACI 90%AC I
θ(0.3269, 2.9347) (0.5365, 2.7251)
α1(0.0, 5.4175) (0.0, 4.7231)
α2(0.0, 2.2795) (0.0, 2.0383)
α3(0.0, 2.7776) (0.0, 2.4200)
α4(0.0, 0.9644) (0.0, 0.8101)
ξ1(0.0, 1131.7) (0.0, 978.45)
ξ2(0.0, 58.645) (0.0, 52.385)
ξ3(0.0, 258.65) (0.0, 225.39)
ξ4(0.0, 43.912) (0.0, 43.912)
γ1(0.0, 0.8653) (0.0, 0.7831)
γ2(0.0249, 0.5314) (0.0656, 0.4907)
γ3(0.013, 0.3478) (0.0399, 0.3209)
γ4(0.4226, 1.2407) (0.4883, 1.1750)
7.2. Data for Warranty Policies
Consider the bivariate real data set from [
33
] consisting of 30 observations of failure
times and warranty servicing times in days for nuclear power plants of the four nuclear
sites in South Korea. Recently, [
35
] analyzed this data using the bivariate exponentiated
additive Weibull distribution. In this subsection, the bivariate real data, summarized in
Table 6, are analyzed to show the performance of the proposed bivariate distribution by
comparing them with some sub-models and the bivariate IWD.
Table 6. Failure times and warranty servicing times for nuclear power plants.
Measurements
Failure times 353.04, 334.72, 80.04, 6.49, 1.34, 467.19, 0.35, 398.86, 1048.23, 829.23,
227.20, 260.14, 14.00, 14.15, 38.96, 30.27, 117.37, 126.27, 56.45, 45.28,
267.31, 615.64, 115.37, 359.76, 412.30, 276.69, 601.04, 1021.01, 192.17, 0.36
Warranty servicing 4.37, 1.91, 2.04, 1.72, 0.29, 1.93, 1.82, 1.77, 9.61, 3.80, 2.86, 0.31, 0.85,
times 2.04, 2.73, 3.63, 2.73, 2.55, 0.72, 3.69, 0.36, 10.63, 11.24, 9.70, 3.31, 4.96,
2.99, 2.36, 1.63, 0.26
The numerical values of the K-S test and the associated p-value of all comparison
uni-variate distributions (marginal functions) are summarized in Table 7for warranty
policy data. It is evident from Table 7that the marginal distributions for all comparison
distributions fit all univariate data sets well based on the p-values. Figure 6shows the Q-Q
plots for two data sets of the warranty policies data. From Figure 6, it is evident that the
marginal distribution of MVMDD fits both data sets of the warranty policies data well.
The MLEs for the unknown parameters of MVMDD, MVDD, MVBIIID, MVMBIIID,
and MVIWD are obtained when the multivariate warranty policies data are used. Table 8
shows the values of the MLEs for MVMDD, MVDD, MVBIIID, MVMBIIID, and MVIWD,
and the associated values for AIC. Based on the values of the AICs for MVMDD, MVDD
MVBIIID, MVMBIIID, and MVIWD, it is evident that MVMDD and MVMBIIID have the
lowest values and are considered better than other models for the warranty policy data.
Additionally, the plots of the profile log-likelihood function of the estimated parameters
for the protein consumption data are shown in Figure 7.

Mathematics 2025,13, 1620 21 of 27
Table 7. K-S test distance with the associated p-value for the warranty policy data.
Uni-Variate Model Data K-S p-Value
MDD Failure times 0.0580 0.9999
Warranty servicing 0.1029 0.9087
DD Failure times 0.0671 0.9993
Warranty servicing 0.1004 0.9231
BIIID Failure times 0.2019 0.1733
Warranty servicing 0.1785 0.2947
MBIIID Failure times 0.0471 0.9999
Warranty servicing 0.1149 0.8231
IWD Failure times 0.2189 0.1127
Warranty servicing 0.2178 0.1162
Figure 6. Q-Q plot for failure times and warranty servicing times in days of the warranty policy data.
Figure 7. The log-likelihood function of estimated parameters of MVMDD for the warranty pol-
icy data.

Mathematics 2025,13, 1620 22 of 27
Table 8. The estimates of the unknown parameters of MVMDD, MVDD, MVBIIID, MVMBIIID,
and MVIWD with the associated AICs for the warranty policy data.
Parameters MVMDD MVDD MVBIIID MVMBIIID MVIWD
ˆ
θ1.2674 0.6852 2.8541 0.2018 2.0508
ˆ
α10.3823 0.8720 0.3801 6.7665 0.3905
ˆ
α21.0443 1.9513 1.3011 0.5389 0.9647
ˆ
ξ19.1664 103.04 1 1 3.6936
ˆ
ξ23.2762 8.8347 1 1 1.2767
ˆ
γ10.0031 0 0 0.0032 -
ˆ
γ20.2234 0 0 0.3712 -
AIC 523.20 535.74 566.84 523.88 558.47
The 95% and 90% ACIs using the asymptotic variance–covariance matrix, given in
Box 2, are computed for the unknown parameters of MVMDD for warranty policy data
and are reported in Table 9.
Table 9. The 95% and 90% ACIs for the unknown parameters of MVMDD for the warranty policy data.
Parameters 95%ACI 90%AC I
θ(0.0, 2.6387) (0.1165, 2.4183)
α1(0.1129, 0.6518) (0.1562, 0.6085)
α2(0.2292, 1.8594) (0.3602, 1.7284)
ξ1(0.0, 31.082) (0.0, 27.5596)
ξ2(0.0, 9.1585) (0.0, 8.2131)
γ1(0.0012, 0.0049) (0.0015, 0.0046)
γ2(0.0249, 0.4942) (0.0656, 0.4507)
8. Conclusions
In this paper, we introduced MVMDD and its sub-models with MDD, DD, MBIIID, and
BIIID as their respective marginals. The proposed model is highly flexible, allowing it to be
derived in two distinct ways for both the PDF and the CDF. We explored various properties
of the model in detail, providing a thorough understanding of its characteristics. While
the MLEs are computationally intensive, we employed a numerical method to facilitate
their estimation. To illustrate the applicability of the model, we analyzed multivariate
protein consumption in a data set of twenty-five European countries and warranty policy
data, consisting of 30 observations of failure times and warranty servicing times in days for
nuclear power plants of the four nuclear sites in South Korea. The results demonstrated
that the proposed model fits the two data sets well, outperforming its sub-models and
MVIWD in terms of the AIC value. This indicates that the proposed model provided a
better fit for these particular data sets. Overall, the paper showcases the flexibility and
practical utility of the model for real-world data analysis. The proposed model offered a
promising approach for modeling multivariate data with specific marginal distributions.
Author Contributions: Conceptualization, N.A., K.S.S. and H.M.M.R.; Methodology, N.A., K.S.S.
and H.M.M.R.; Software, H.M.M.R.; Validation, N.A.; Formal analysis, K.S.S. and H.M.M.R.; Data
curation, K.S.S. and H.M.M.R.; Writing—original draft, K.S.S. and H.M.M.R.; Writing—review &
editing, N.A.; Supervision, N.A.; Project administration, N.A. All authors have read and agreed to
the published version of the manuscript.
Funding: This research was funded by Researchers Supporting Project number (PNURSP2025R523),
Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
Data Availability Statement: Data available within the article.

Mathematics 2025,13, 1620 23 of 27
Acknowledgments: Princess Nourah bint Abdulrahman University Researchers Supporting Project
number (PNURSP2025R523), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding
the present study.
Abbreviations
The following abbreviations are used in this manuscript:
CDF Cumulative distribution function
PDF Probability density function
AIC Akaike Information Criterion
IWD Inverse Weibull distribution
GIWD Generalized inverse Weibull distribution
MGIWD Modified generalized inverse Weibull distribution
DD Dagum distribution
MDD Modified Dagum distribution
MVDD Multivariate Dagum distribution
MVMDD Multivariate modified Dagum distribution
BIIID Burr Type-III distribution
MBIIID Modified Burr Type-III distribution
MVBIIID Multivariate Burr Type-III distribution
MVBIIID Multivariate Modified Burr Type-III distribution
MVLD Multivariate Logistic Distribution
MVGED Multivariate Generalized exponential distribution
MVIWD Multivariate inverse Weibull distribution
GB2 Generalized beta distribution of the second kind
MVGED Absolutely continuous multivariate generalized exponential distribution
PLOD Positively lower orthant dependent
LTD Left tail decreasing
MLE Maximum likelihood estimation
Q-Q plot Quantile-quantile plot
Appendix A
The second derivatives of log-likelihood function (27) are derived as
∂2ℓ
∂θ2=
nΓ(θ)ψ(0)(θ)Γ(k+θ)ψ(0)(k+θ)
Γ(θ)−ψ(0)(θ)Γ(k+θ)
Γ(θ)
Γ(k+θ)−
nΓ(θ)ψ(0)(k+θ)Γ(k+θ)ψ(0)(k+θ)
Γ(θ)−ψ(0)(θ)Γ(k+θ)
Γ(θ)
Γ(k+θ)
+
nΓ(θ)ψ(0)(θ)2Γ(k+θ)
Γ(θ)−2ψ(0)(θ)Γ(k+θ)ψ(0)(k+θ)
Γ(θ)+Γ(k+θ)ψ(0)(k+θ)2
Γ(θ)+Γ(k+θ)ψ(1)(k+θ)
Γ(θ)−ψ(1)(θ)Γ(k+θ)
Γ(θ)
Γ(k+θ),
∂2ℓ
∂α2
i
=
k
∑
s=1
n
∑
j=1
−δi,s2
γsxjs +αs2−(θ+k)
n
∑
j=1 ∑k
s=1ξsδi,s2x−αs
js eγs(−xjs )log2xjs
∑k
i=1ξix−αi
ij eγi(−xi j)+1
−∑k
s=1ξsδi,sx−αs
js −eγs(−xjs )logxjs 
∑k
i=1ξix−αi
ij eγi(−xi j)+12
∑k
m=1ξmδi,mx−αm
jm −eγm(−xjm )logxjm 
∑k
i=1ξix−αi
ij eγi(−xi j)+12!,

Mathematics 2025,13, 1620 24 of 27
∂2ℓ
∂ξ2
i
=nk
∑
s=1
−δi,s2
ξ2
s
−(θ+k)
n
∑
j=1
−∑k
s=1δi,sx−αs
js eγs(−xjs )∑k
m=1δi,mx−αm
jm eγm(−xjm )
∑k
i=1ξix−αi
ij eγi(−xi j)+12,
∂2ℓ
∂γ2
i
=
k
∑
s=1
n
∑
j=1
−x2
jsδi,s2
γsxjs +αs2−(θ+k)
n
∑
j=1 ∑k
s=1ξsδi,s2x2−αs
js eγs(−xjs )
∑k
i=1ξix−αi
ij eγi(−xi j)+1
−∑k
s=1ξsδi,sx1−αs
js −eγs(−xjs )∑k
m=1ξmδi,mx1−αm
jm −eγm(−xjm )
∑k
i=1ξix−αi
ij eγi(−xi j)+12!,
∂2ℓ
∂αi∂θ =−
n
∑
j=1
∑k
s=1ξsδi,sx−αs
jK[1]−eγs(−xjs )logxjs 
∑k
i=1ξix−αi
ij eγi(−xi j)+1
,
∂2ℓ
∂αi∂ξi=− (θ+k)
n
∑
j=1 ∑k
s=1δi,s2x−αs
js −eγs(−xjs )logxjs 
∑k
i=1ξix−αi
ij eγi(−xi j)+1
−∑k
m=1δi,mx−αm
jm eγm(−xjm )∑k
s=1ξsδi,sx−αs
js −eγs(−xjs )logxjs 
∑k
i=1ξix−αi
ij eγi(−xi j)+12!!,
∂2ℓ
∂αi∂γi=
k
∑
s=1
n
∑
j=1
−xjsδi,s2
γsxjs +αs2−(θ+k)
n
∑
j=1 ∑k
s=1ξsδi,s2x1−αs
js eγs(−xjs )logxjs 
∑k
i=1ξix−αi
ij eγi(−xi j)+1
−∑k
s=1ξsδi,sx−αs
js −eγs(−xjs )logxjs 
∑k
i=1ξix−αi
ij eγi(−xi j)+12
∑k
m=1ξmδi,mx1−αm
jm −eγm(−xjm )
∑k
i=1ξix−αi
ij eγi(−xi j)+12!,
∂2ℓ
∂ξi∂γi=− (θ+k)
n
∑
j=1 ∑k
s=1δi,s2x1−αs
js −eγs(−xjs )
∑k
i=1ξix−αi
ij eγi(−xi j)+1
−∑k
s=1δi,sx−αs
js eγs(−xjs )∑k
m=1ξmδi,mx1−αm
jm −eγm(−xjm )
∑k
i=1ξix−αi
ij eγi(−xi j)+12!!,
∂2ℓ
∂ξi∂θ =−
n
∑
j=1
∑k
s=1δi,sx−αs
js eγs(−xjs )
∑k
i=1ξix−αi
ij eγi(−xi j)+1
,
∂2ℓ
∂θ∂γi=−
n
∑
j=1
∑k
s=1ξsδi,sx1−αs
js −eγs(−xjs )
∑k
i=1ξix−αi
ij eγi(−xi j)+1
.

Mathematics 2025,13, 1620 25 of 27
The calculated variance covariance matrix for the protein consumption data
0.443 −0.184 −0.098 −0.093 −0.037 −116.947 −9.323 −30.646 −7.184 0.007 0.004 −0.001 −0.016
−0.184 4.860 0.039 0.044 0.010 1007.020 3.952 16.281 3.190 −0.552 −0.001 0.003 0.010
−0.098 0.039 0.586 0.023 0.052 26.527 10.347 8.164 2.542 −0.001 −0.088 0.001 0.001
−0.093 0.044 0.023 1.289 0.012 29.361 2.450 109.087 1.630 0.000 0.001 −0.090 0.003
−0.037 0.010 0.052 0.012 0.240 8.675 1.735 3.495 0.132 0.000 −0.005 0.000 −0.082
−116.947 1007.020 26.527 29.361 8.675 236,588.000 2646.000 9899.980 2075.340 −104.521 −0.678 0.892 5.596
−9.323 3.952 10.347 2.450 1.735 2646.000 395.027 757.382 185.996 −0.134 −0.954 0.026 0.341
−30.646 16.281 8.164 109.087 3.495 9899.980 757.382 11,145.500 556.681 −0.526 −0.150 −6.483 1.257
−7.184 3.190 2.542 1.630 0.132 2075.340 185.996 556.681 188.409 −0.119 −0.141 0.031 1.238
0.007 −0.552 −0.001 0.000 0.000 −104.521 −0.134 −0.526 −0.119 0.068 0.000 0.000 0.000
0.004 −0.001 −0.088 0.001 −0.005 −0.678 −0.954 −0.150 −0.141 0.000 0.017 0.000 0.001
−0.001 0.003 0.001 −0.090 0.000 0.892 0.026 −6.483 0.031 0.000 0.000 0.007 0.000
−0.016 0.010 0.001 0.003 −0.082 5.596 0.341 1.257 1.238 0.000 0.001 0.000 0.044
The calculated variance covariance matrix for the warranty policies data
0.4895 −0.06 −0.1857 −7.1968 −1.9365 1.8 ×10−50.0239
−0.06 0.0189 0.0299 1.236 0.2524 −0.0001 −0.0042
−0.1857 0.0299 0.1729 3.0543 0.6887 2.4 ×10−5−0.0416
−7.1968 1.236 3.0543 125.02 30.2153 −0.001 −0.4013
−1.9365 0.2524 0.6887 30.2153 9.0072 −0.0001 −0.0206
1.8 ×10−5−0.0001 2.4 ×10−5−0.001 −0.0001 8.9 ×10−75.2 ×10−6
0.0239 −0.0042 −0.0416 −0.4013 −0.0206 5.2 ×10−60.0191
Appendix B
The code of MLE for the the warranty policies data in Mathematica 11.
• Set the the warranty policies data as follows:
x1 = {353.04, . . ., 0.36}; x2 = {4.37, . . ., 0.26};
• Set the log-likelihood function
n = Length[x1]
L=nlogφ1φ2Γ(θ+2)
Γ(θ)+
n
∑
i=1
logx1[[i]]−α1−1x2[[i]]−α2−1−
n
∑
i=1
γ1x1[[i]]
−
n
∑
i=1
γ2x2[[i]] +
n
∑
i=1
log(α1+γ1x1[[i]]) +
n
∑
i=1
log(α2+γ2x2[[i]])
−
n
∑
i=1
logφ1x1[[i]]−α1e−γ1x1[[i]] +φ2x2[[i]]−α2e−γ2x2[[i]] +1θ+2;
• Use FindRoot in Wol f ra mMat hemat ica 11 to solve seven normal equations
solution =FindRoot"∂L
∂θ =0, ∂L
∂α1=0, ∂L
∂α2=0, ∂L
∂φ1=0, ∂L
∂φ2=0, ∂L
∂γ1=0, ∂L
∂γ2=0,
Transpose[{{θ,α1, α2, φ1, φ2, γ1, γ2},RandomReal[guessvalue,numbero f p aramet ers]}],
MaxIterations →100000#;
• To show the convergence of the solution, the following steps are required:
grd =Grad[L,{θ,α1, α2, α3, α4, φ1, φ2, φ3, φ4, γ1, γ2, γ3, γ4}];
grd/.solution

Mathematics 2025,13, 1620 26 of 27
•
To obtain the asymptotic variance–covariance matrix, the following steps are required:
hess =D[L,{{θ,α1, α2, α3, α4, φ1, φ2, φ3, φ4, γ1, γ2, γ3, γ4}, 2}];
m=−Inverse[hess/.sol];
MatrixForm[m]
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