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Citation: Guan, Q.; Jiang, P.; Liu, G.
Some New Bivariate Properties and
Characterizations Under Archimedean
Copula. Mathematics 2024,12, 3714.
https://doi.org/10.3390/
math12233714
Academic Editor: Wanyang Dai
Received: 30 October 2024
Revised: 21 November 2024
Accepted: 22 November 2024
Published: 26 November 2024
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mathematics
Article
Some New Bivariate Properties and Characterizations Under
Archimedean Copula
Qingyuan Guan 1,2,3, Peihua Jiang 4, * and Guangyu Liu 4
1School of Mathematics and Computer Science, Wuyi University, Wuyishan 354300, China;
qingyuanguan@wuyiu.edu.cn or qingyuan guan@163.com
2Key Laboratory of Cognitive Computing and Intelligent Information Processing of Fujian Education
Institutions, Wuyishan 354300, China
3Fujian Key Laboratory of Big Data Application and Intellectualization for Tea Industry,
Wuyishan 354300, China
4School of Mathematics-Physics and Finance, Anhui Polytechnic University, Wuhu 241000, China;
guangyuliu666@126.com
*Correspondence: jiangph@ahpu.edu.cn or jiangph2017@163.com
Abstract: This paper considers comparing properties and characterizations of the bivariate functions
under Archimedean copula. It is shown that some results of the usual stochastic order for the bivariate
functions in the independent case are generalized to the Archimedean copula-linked dependent case,
and we also derive some characterizations of different bivariate functions composed by Archimedean
copula-linked dependent random variables. These results generalize some existing results in the
literature and bring conclusions closer to reality. Two applications in scheduling problems are also
provided to illustrate the main results.
Keywords: usual stochastic order; characterizations; bivariate functions; Archimedean copula
MSC: 62E15; 62N05; 62G30
1. Introduction
Various concepts of stochastic ordering provide a variety of useful comparisons in
insurance actuarial science, reliability, etc. Many researchers have studied the nature of
the (reversed) hazard rate and likelihood ratio order. Lynch et al.
[1]
examined some
closure properties of hazard rate order, while Oliveira and Torrado
[2]
showed the char-
acteristics and closed properties of a decreasing proportional reversed hazard rate class.
Boland et al.
[3]
presented the application of hazard rate order in reliability and [
4
] dis-
cussed the reliability application of the reversed hazard order. In the literature [
5
–
12
],
the stochastic comparisons of series and parallel systems with independent components
have been effectively investigated through the smallest and the largest order statistics in
the sense of (reversed) hazard rate order. Esna-Ashari et al.
[13]
compared generalized
order statistics in the sense of hazard rate order and reversed hazard order. Also, some
scholars studied the likelihood ratio ordering properties of the smallest and the largest
order statistics; see [
14
,
15
]. Furthermore, Barmalzan et al.
[16]
established the likelihood
ratio order properties of the smallest claim amounts of two independent heterogeneous
Weibull distributions.
On the other hand, the past several decades have witnessed many investigations
on the bivariate characterization. Muraleedharan and Unnikrishnan
[17]
discussed the
characterization of Gumbel’s bivariate exponential distribution based on the properties of
the conditional moments. Shanthikumar and Yao
[18]
showed the bivariate characterization
in the sense of the likelihood ratio order, the hazard rate order and the usual stochastic order.
Meanwhile, based on collections of pairs of bivariate functions, Righter and Shanthikumar
Mathematics 2024,12, 3714. https://doi.org/10.3390/math12233714 https://www.mdpi.com/journal/mathematics
Mathematics 2024,12, 3714 2 of 11
[19]
extended the result in [
18
]. Thomas and Veena
[20]
derived some properties of
concomitants of record values which characterize the generalized class of distributions.
Noughabi and Kayid
[21]
proposed and studied bivariate
α−
quantile residual life measure.
Most of the existing research that compares random variables assumes that they are
all independent. However, the operating environment of such technical systems is often
affected by many factors, such as operating conditions, environmental conditions and the
stress factors on the components. For this reason, it would be prudent to consider dependent
random variables. Archimedean copulas are a type of multivariate probability distribution
used to model the dependence relationship among random variables. Moreover, they also
incorporate the independence copula as a special case. For a comprehensive review of
copulas and their applications, see [22–25].
Under Archimedean copulas, Barmalzan et al.
[26]
proved the comparison properties
of the hazard rate order and reversed hazard order, which is the extension of [
8
]. Li and
Fang
[27]
discussed stochastic orderings of the extremal values and their adjacent order
statistics in the sense of both likelihood ratio order and (reversed) hazard rate order, while
Mesfioui et al.
[28]
extended the result of [
27
] to the heterogeneous random variables linked
by an Archimedean copula. Ariyafar et al.
[29]
compared the aggregation and minimum
of these portfolios with respect to the Laplace transform order according to Archimedean
copulas. Fang et al.
[30]
presented the stochastic comparison results for the largest and
smallest order statistics from dependent Gaussian variables with Archimedean copula
under simple tree order restrictions. Lu et al.
[31]
used the asymmetric Archimedean
copula to characterize the multiple uncertainties of a W-PV-H system. Amini et al.
[32]
considered two
k
-out-of-
n
systems comprising heterogeneous dependent components
under random shocks with an Archimedean copula. Recently, Guan and Wang
[33]
proved stochastic comparison properties of convex and increasing convex order under
Archimedean copulas. We find that for dependent random variables, fewer researchers have
considered the usual stochastic order and the characterization of bivariate functions based
on Archimedean copula. The objective of this study is to generalize the usual stochastic
order and the characterization of bivariate functions composed by Archimedean copula-
linked dependent random variables. The results on stochastic orders under Archimedean
copulas mentioned above motivate us to consider the problems discussed in this work.
The rest of this paper is organized as follows. In Section 2, some basic definitions and
notation are given which are most pertinent to the results established in the subsequent
sections. In Section 3, some useful lemmas are provided which establish the relationship
between independent and Archimedean copula-linked dependent random variables. In Sec-
tion 4, some new results regarding the usual stochastic order and characterization among
bivariate functions of dependent random variables are obtained, which extend some known
results in the literature and bring conclusions closer to reality. In Section 5, two applications
are given. Finally, we provide some conclusions in Section 6.
2. Preliminaries
Throughout this paper, we denote
R= (−∞
,
∞)
,
R+= [
0,
∞)
,
Rn={(z1
,
. . .
,
zn):
zi∈(−∞
,
∞)
, for all
i}
. The terms “increasing" and “decreasing" are used to denote
“monotone non-decreasing" and “monotone non-increasing", respectively. We suppose that
all random variables are defined on a common probability space
(Ω
,
F
,P
)
and that all
expectations exist wherever they are used.
2.1. Stochastic Orders
Definition 1. Suppose that random variables
X
and
Y
have distribution functions
FX
and
FY
,and
survival functions FX=1−FXand FY=1−FY, respectively.
(i)
If
FX(x)≤FY(x)
for all
x∈R
,then we say that
X
is smaller than
Y
in the usual stochastic
order (denoted by X ≤st Y);
(ii)
If
FY(x)/FX(x)
is increasing in
x∈R+
, then we say that
X
is smaller than
Y
in the hazard
rate order (denoted by X ≤hr Y);
Mathematics 2024,12, 3714 3 of 11
(iii) If
FY(x)/FX(x)
is increasing in
x∈R+
, then we say that
X
is smaller than
Y
in the reversed
hazard order (denoted by X ≤rh Y);
(iv) If
fY(x)/fX(x)
is increasing in
x∈R+
, then we say that
X
is smaller than
Y
in the likelihood
ratio order (denoted by X ≤l r Y).
It is well known that the likelihood ratio order implies the hazard rate order and the reversed
hazard order, and the hazard rate order or the reversed hazard order implies the usual stochastic
order. For more details on stochastic orderings and their applications, one may refer to [34,35].
2.2. Copulas
In this section, we present some basic notions of copulas.
Let
F
be an
n
-dimensional distribution function with marginal distributions
F1
,
. . .
,
Fn
.
A copula associated with Fis a distribution function C:[0, 1]n→[0, 1]satisfying
F(x1, . . . , xn)=C(F1(x1), . . . , Fn(xn)).
Similarly, a survival copula associated with
F
is a survival function
C:[
0, 1
]n→
[
0, 1
]
satisfying
F(x1, . . . , xn)=CF1(x1), . . . , Fn(xn)
. For comprehensive discussions on
copulas, one may refer to [36–38].
Definition 2 ([
36
]).If for all
t∈(a
,
b)
, all derivatives of a function
h(t)
exist and satisfy
(−
1
)kh(k)(t)≥
0,
k∈ {
0, 1,
. . .}
, where
h(k)(·)
denotes the
k
th derivative of
h(·)
; then,
h(·)
is
said to be completely monotone on an interval (a,b),a,b∈R.
Thus, if
ϕ−1
is a completely monotone function, there exists a distribution function
Lϕ−1
such that
ϕ−1(x) = Z∞
0exp(−αx)dLϕ−1(α).
Definition 3 ([
37
]).For a completely monotone function
ϕ−1:[
0,
+∞)→[
0, 1
]
with
ϕ−1(
0
) =
1
and
limx→+∞ϕ−1(x) =
0, then
Cϕ(u) = ϕ−1(ϕ(u1)+· · · +ϕ(un))
, for all
ui∈(
0, 1
)
,
i∈In
,
is called an Archimedean copula with strict generator ϕ, where ϕ−1(u)is the pseudo-inverse of ϕ.
For example,
C(u1
,
. . .
,
un) = min(u1
,
. . .
,
un)
is an
n
-copula, and the Gumbel copula is a
special case of strict Archimedean copulas with generator
ϕ(t) = (−ln t)θ
,
θ≥
1. For ease of nota-
tion, we denote the vector
FX1(x1), . . . , FXn(xn)
by
FXn(xn)
and the vector
FX1(x1), . . . , FXn(xn)
by FXn(xn).
3. Some Useful Lemmas
For establishing the main results in the following section, the following lemmas
are required.
Lemma 1 ([
33
]).Let
CFUn(un)
be the joint survival function of
(U1
,
. . .
,
Un)
, where
C
is
the Archimedean copula with generator
ϕ
. For
i=
1,
. . .
,
n
, let
Ui(α)
’s be independent random
variables with survival functions
exp−αϕFUi
. Then, for all
α>
0and all continuous
functions g :Rn→R, we have
E[g(U1, . . . , Un)] =Z∞
0E[g(U1(α), . . . , Un(α))]dLϕ−1(α).
Remark 1. In fact, the above conclusion only requires
g
to be a measurable function but not a
continuous function. This is because the process of proving Lemma 1mainly uses the Fubini theorem,
which holds as long as g is a measurable function.
Lemma 2. Let
U
and
V
be two random variables. Assume that
ϕ:(
0, 1
]→R+
is a monotone
decreasing function with
ϕ(
1
) =
0,
ϕ(
0
) = ∞
. For all
α>
0, let the survival functions of
U(α)
and
V(α)
be given by
exp−αϕFU
and
exp−αϕFV
, respectively. Then,
U(α)≤hr V(α)
for all α>0if and only if ϕFV(t)−ϕFU(t)is decreasing in t ∈R+.
Mathematics 2024,12, 3714 4 of 11
Proof. Note that for α>0, we have
U(α)≤hr V(α)⇐⇒ FV(α)(t)
FU(α)(t)is increasing in t ∈R+,
⇐⇒ exp{−α[ϕ(FV(t)) −ϕ(FU(t))]}is increasing in t ∈R+,
⇐⇒ ϕFV(t)−ϕFU(t)is decreasing in t ∈R+,
which is the desired result.
Lemma 3. Suppose
U
and
V
are two random variables. Assume that
ϕ:(
0, 1
]→R+
is a monotone
decreasing function with
ϕ(
1
) =
0,
ϕ(
0
) = ∞
. For all
α>
0, let the survival functions of
U(α)
and
V(α)
be given by
exp−αϕFU
and
exp−αϕFV
, respectively. Then,
U(α)≤rh V(α)
for all α>0if and only if ϕ(FV(t))−ϕ(FU(t))is decreasing in t ∈R+.
Proof. For all α>0,
U(α)≤rh V(α)⇐⇒ FV(α)(t)
FU(α)(t)is increasing in t∈R+
⇐⇒ exp{−α[ϕ(FV(t)) −ϕ(FU(t))]}is increasing in t∈R+
⇐⇒ ϕ(FV(t))−ϕ(FU(t))is decreasing in t∈R+.
Lemma 4. Suppose
U
and
V
are two random variables. Assume that
ϕ:(
0, 1
]→R+
is a
monotone decreasing function with ϕ(1) = 0,ϕ(0) = ∞.
(i)
For all
α>
0, let the survival functions of
U(α)
and
V(α)
be given by
exp−αϕFU
and
exp−αϕFV, respectively. Then, U(α)≤l r V(α)for all α>0if and only if
ϕ′(FV)
ϕ′(FU)
fV
fU
exp{ϕFU−ϕFV}
is increasing in t ∈R+.
(ii)
For all
α>
0, let the distribution functions of
U(α)
and
V(α)
be given by
exp{−αϕ(FU)}
and exp{−αϕ(FV)}, respectively. Then, U(α)≤lr V(α)for all α>0if and only if
ϕ′(FV)
ϕ′(FU)
fV
fU
exp{ϕ(FU)−ϕ(FV)}
is increasing in t ∈R+.
Proof.
(i)
For all α>0,
U(α)≤lr V(α)⇐⇒ fV(α)(t)
fU(α)(t)is increasing in t∈R+
⇐⇒ αϕ′(FV)fVexp{−αϕ(FV)}
αϕ′(FU)fUexp{−αϕ(FU)}is increasing in t∈R+
⇐⇒ ϕ′(FV)
ϕ′(FU)
fV
fU
exp{ϕFU−ϕFV}is increasing in t∈R+.
(ii)
The proof is similar to that of (i).
Mathematics 2024,12, 3714 5 of 11
4. Main Results
The following Theorem 1, which is a generalization of Theorem 1.
B
.9 of [
35
], gives the
usual stochastic ordering property of the bivariate functions under Archimedean copula.
Theorem 1. Suppose
CFU(u),FV(v)
is the joint survival function of
(U
,
V)
, where
C
is the
Archimedean copula with generator
ϕ
such that
ϕ−1
is a completely monotone function. Then,
ϕFV(t)−ϕFU(t)is decreasing in t ∈R+if and only if, for all g ∈ Ghr,
g(U,V)≤st g(V,U).
where
Ghr =g:R2→R:g(u
,
v)
is increasing in
u
, for each
v
, on
{u≥v}
, and is decreasing
in v, for each u, on {v≥u}}.
Proof.
For all
α>
0, let the random variables
U(α)
and
V(α)
be independent, and let the
survival function of
U(α)(V(α))
be
exp{−αϕFU}exp{−αϕFV}
, respectively. Note
that the condition
ϕFV(t)−ϕFU(t)
is decreasing in
t∈R+
; using Lemma 2, we have
U(α)≤hr V(α). Then, it can be obtained from Theorem 1.B.9 of [35] that for all g∈ Ghr,
g(U(α),V(α)) ≤st g(V(α),U(α)).
This means that for all increasing functions φ, we have
E[φ(g(U(α),V(α)))] ≤E[φ(g(V(α),U(α)))]. (1)
In addition, according to Lemma 1, we have
E[φ(g(U,V))] =Z∞
0E[φ(g(U(α),V(α)))]dLϕ−1(α), (2)
and
E[φ(g(V,U))] =Z∞
0E[φ(g(V(α),U(α)))]dLϕ−1(α). (3)
Based on (1)–(3), we can obtain
E[φ(g(U,V))] ≤E[φ(g(V,U))].
Thus, we have g(U,V)≤st g(V,U)for all g∈ Ghr.
Now, suppose for all g∈ Ghr,
g(U,V)≤st g(V,U).
Select a
u
and a
v
such that
u≥v
. Let
g(x
,
y) = I{x≥u,y≥v}
, where
IA
denotes the
indicator function of the set A. It is easy to see that g(x,y)is increasing in x. Additionally,
for fixed
x
and
y
, such that
y≥x
, we have that
g(x
,
y) =
1 if
x≥u
and
g(x
,
y) =
0 if
x<u
.
Therefore, g∈ Ghr. Hence, we have
Eg(V,U)≥Eg(U,V), (4)
whenever u≥v. Note that
Eg(V,U) = EI{V≥u,U≥v}=ϕ−1ϕFV(u)+ϕFU(v), (5)
and
Eg(U,V) = EI{U≥u,V≥v}=ϕ−1ϕFU(u)+ϕFV(v), (6)
Mathematics 2024,12, 3714 6 of 11
where
FU
and
FV
are the survival functions of
U
and
V
, respectively. Note that
ϕ−1
is a
decreasing function; upon combining (4)–(6), we now have
ϕFV(u)−ϕFU(u)≤ϕFV(v)−ϕFU(v),
which means that
ϕFV(t)−ϕFU(t)
is decreasing in
t∈R+
. Hence, the proof of the
theorem is completed.
Remark 2. Theorem 1shows a necessary and sufficient condition for bivariate functions consisting
of two Archimedean copula-linked dependent random variables to maintain the usual stochastic
order after exchanging the order of variables.
Theorem 2below generalizes Theorem 1.
B
.47 of [
35
] to an Archimedean copula-linked
dependent case. The result shows another necessary and sufficient condition for bivariate
functions to preserve a usual stochastic order under Archimedean copula.
Theorem 2. Suppose
CFU(u),FV(v)
is the joint survival function of
(U
,
V)
, where
C
is the
Archimedean copula with generator
ϕ
such that
ϕ−1
is a completely monotone function. For all
g∈ Grh,ϕ(FV(t))−ϕ(FU(t))is decreasing in t ∈R+if and only if
g(U,V)≤st g(V,U),
where
Grh =g:R2→R:g(u
,
v)
is increasing in
u
, for each
v
, on
{u≤v}
, and is decreasing
in v, for each u, on {v≤u}}.
Proof.
For all
α>
0, let
U(α)
and
V(α)
be independent random variables, and let the
survival functions of
U(α)
and
V(α)
be given by
exp−αϕFU
and
exp−αϕFV
,
respectively. Note the condition that
ϕFV(t)−ϕFU(t)
is decreasing in
t∈R+
; then,
it can be obtained from Lemma 3that
U(α)≤rh V(α)
. Then, according to Theorem 1.
B
.47
of [35], for all g∈ Grh , we have
g(U(α),V(α)) ≤st g(V(α),U(α)),
which means that for any increasing function φ,
E[φ(g(U(α),V(α)))] ≤E[φ(g(V(α),U(α)))]. (7)
Moreover, by Lemma 1, we have
E[φ(g(U,V))] =Z∞
0E[φ(g(U(α),V(α)))]dLϕ−1(α), (8)
and
E[φ(g(V,U))] =Z∞
0E[φ(g(V(α),U(α)))]dLϕ−1(α). (9)
Using (7)–(9), we have
E[φ(g(U,V))] ≤E[φ(g(V,U))],
which means
g(U,V)≤st g(V,U).
Now, suppose for all g∈ Grh ,
g(U,V)≤st g(V,U).
Mathematics 2024,12, 3714 7 of 11
Select a
u
and a
v
such that
u≤v
. Let
g(x
,
y) =
1
−I{x≤u,y≤v}
, where
IA
denotes the
indicator function of the set
A
. It is easy to see that
g(x
,
y)
is increasing in
x
. What is more,
for fixed
x
and
y
, such that
y≤x
, we have that
g(x
,
y) =
1 if
x≥u
and
g(x
,
y) =
0 if
x<u
.
Therefore, g∈ Grh. Hence, we have
Eg(V,U)≥Eg(U,V), (10)
whenever u≤v. Note that
Eg(V,U) = 1−EI{V≤u,U≤v}=1−ϕ−1(ϕ(FV(u))+ϕ(FU(v))), (11)
and
Eg(U,V) = 1−EI{U≤u,V≤v}=1−ϕ−1(ϕ(FU(u))+ϕ(FV(v))), (12)
where
FU
and
FV
are the distribution functions of
U
and
V
, respectively. Note that
ϕ−1
is a
decreasing function; combining (10)–(12), we have
ϕ(FV(u))−ϕ(FU(u))≤ϕ(FV(v))−ϕ(FU(v)),
which means that
ϕ(FV(t))−ϕ(FU(t))
is decreasing in
t∈R+
. Thus, the required re-
sult follows.
The following Theorem 3generalizes Theorem 1.
C
.20 of [
35
] to an Archimedean copula-
linked dependent case. This result shows a sufficient condition for bivariate functions to
preserve the usual stochastic order under Archimedean copula.
Theorem 3. Let
CFU(u),FV(v)
be the joint survival function of
(U
,
V)
, where
C
is the Archimedean
copula with generator
ϕ
such that
ϕ−1
is a completely monotone function. For all
g∈ Glr =
g:R2→R:g(u,v)≤g(v,u)whenever u ≤v, if
ϕ′(FV)
ϕ′(FU)
fV
fU
exp{ϕFU−ϕFV}
is increasing in t ∈R+, then g(U,V)≤st g(V,U).
Proof.
For all
α>
0, let the random variables
U(α)
and
V(α)
be independent, and let
the survival functions of
U(α)
and
V(α)
be given by
exp−αϕFU
and
exp−αϕFV
,
respectively. Considering that
ϕ′(FV)
ϕ′(FU)
fV
fUexp{ϕFU−ϕFV}
is increasing in
t∈R+
, then,
by Lemma 4, we have
U(α)≤lr V(α)
. Further, it can be acquired from Theorem 1.
C
.20
of [35] that for all g∈ Glr,
g(U(α),V(α)) ≤st g(V(α),U(α)),
which means for all increasing functions φ, we have
E[φ(g(U(α),V(α)))] ≤E[φ(g(V(α),U(α)))]. (13)
In addition, using Lemma 1, we have
E[φ(g(U,V))] =Z∞
0E[φ(g(U(α),V(α)))]dLϕ−1(α), (14)
and
E[φ(g(V,U))] =Z∞
0E[φ(g(V(α),U(α)))]dLϕ−1(α). (15)
Mathematics 2024,12, 3714 8 of 11
Combining (13)–(15), we obtain
E[φ(g(U,V))] ≤E[φ(g(V,U))].
Therefore, for any g∈ Glr , we have
g(U,V)≤st g(V,U).
This completes the proof.
Similarly, we can prove the following result, which is the generalization of Theo-
rem 1.C.23 of [35] under Archimedean copula. The proof process is omitted for brevity.
Theorem 4. Suppose
CFU(u),FV(v)
is the joint survival function of
(U
,
V)
, where
C
is the
Archimedean copula with generator ϕsuch that ϕ−1is a completely monotone function. If
ϕ′(FV)
ϕ′(FU)
fV
fU
exp{ϕFU−ϕFV}
is increasing in t ∈R+, then
ϕ1(U,V)≤st ϕ2(U,V),
for all
ϕ1
and
ϕ2
, that satisfy
∆ϕ21(u
,
v)≥
0whenever
u≤v
, and
ϕ1(u
,
v)≤ϕ2(v
,
u)
, for all
u
and v, where ∆ϕ21(u,v) = ϕ2(u,v)−ϕ1(u,v).
The following three theorems extend Theorem 1.
B
.10, 1.
B
.48 and 1.
C
.22 of [
35
] to the
Archimedean copula-linked dependent case. These results show three sufficient conditions
for the characterization of different bivariate functions under Archimedean copula.
Theorem 5. Let
CFU(u),FV(v)
be the joint survival function of
(U
,
V)
, where
C
is the
Archimedean copula with generator
ϕ
such that
ϕ−1
is a completely monotone function. If
ϕFV(t)−
ϕFU(t)is decreasing in t ∈R+, then
Eϕ1(U,V)≤Eϕ2(U,V),
for all
ϕ1
and
ϕ2
such that,
∆ϕ21(u
,
v)
increases in
v
on
{v≥u}
, and such that
∆ϕ21(u
,
v)≥
−∆ϕ21(v,u)whenever u ≤v, where ∆ϕ21(u,v) = ϕ2(u,v)−ϕ1(u,v).
Proof.
For all
α>
0, let
U(α)
and
V(α)
be independent random variables, and let the
survival function of
U(α) [V(α)]
be
exp{−αϕFU}[exp{−αϕFV}]
, respectively. Since
ϕFV(t)−ϕFU(t)
is decreasing in
t∈R+
, by Lemma 2, we have
U(α)≤hr V(α)
.
Further, it then can be obtained from Theorem 1.B.10 of [35] that
Eϕ1(U(α),V(α)) ≤Eϕ2(U(α),V(α)). (16)
Furthermore, by Lemma 1, we obtain
Eϕ1(U,V) = Z∞
0Eϕ1(U(α),V(α))dLϕ−1(α), (17)
and
Eϕ2(U,V) = Z∞
0Eϕ2(U(α),V(α))dLϕ−1(α). (18)
Using (16)–(18), we obtain
Eϕ1(U,V)≤Eϕ2(U,V),
Mathematics 2024,12, 3714 9 of 11
and completing the proof.
Remark 3. Theorem 5indicates a sufficient condition for the expectation of different bivariate
functions composed of two Archimedean copula-linked dependent random variables to maintain the
size relationship.
Theorem 6. Suppose
CFU(u),FV(v)
is the joint survival function of
(U
,
V)
, where
C
is
the Archimedean copula with generator
ϕ
such that
ϕ−1
is a completely monotone function. If
ϕ(FV(t))−ϕ(FU(t))is decreasing in t ∈R+, then
Eϕ1(U,V)≤Eϕ2(U,V),
for all
ϕ1
and
ϕ2
such that for each
v
,
∆ϕ21(u
,
v)
decreases in
u
on
{u≤v}
and such that
∆ϕ21(u,v)≥ −∆ϕ21 (v,u)whenever u ≤v, where ∆ϕ21(u,v) = ϕ2(u,v)−ϕ1(u,v).
Theorem 7. Let
CFU(u),FV(v)
be the joint survival function of
(U
,
V)
, where
C
is the
Archimedean copula with generator
ϕ
such that
ϕ−1
is a completely monotone function. For all
functions
ϕ1
and
ϕ2
,
∆ϕ21(u
,
v)≥
0whenever
u≤v
, and
∆ϕ21(u
,
v)≥ −∆ϕ21(v
,
u)
whenever
u≤v, ∆ϕ21(u,v) = ϕ2(u,v)−ϕ1(u,v), if
ϕ′(FV)
ϕ′(FU)
fV
fU
exp{ϕFU−ϕFV}
is increasing in t ∈R+, then Eϕ1(U,V)≤Eϕ2(U,V).
Remark 4. The proofs of both Theorems 6and 7are similar to the proof of Theorem 5; for the sake
of brevity, they are omitted here.
Remark 5. It should be emphasized that the results in Theorems 3,4and 7still hold when
CFU(u),FV(v)
is replaced by
C(FU(u),FV(v))
and the condition that the increasing function
ϕ′(FV)
ϕ′(FU)
fV
fUexp{ϕFU−ϕFV}is replaced by ϕ′(FV)
ϕ′(FU)
fV
fUexp{ϕ(FU)−ϕ(FV)}.
5. Scheduling Application
In this section, two potential applications of our results are presented. Consider the
following scheduling problem. In order to minimize the total cost,
n
jobs are scheduled on
one machine. The processing time
Xi
of the
i
th job and the processing time
Xj
of the
j
th job
are linked by an Archimedean copula with generator
ϕ
.
Di
denotes the completion time of
job
i
,
gi
denotes its cost function; then,
TC =∑n
i=1gi(Di)
denotes the total cost. If
gi−gj
is
increasing,
gi
is said to be steeper than
gj
, which is written
gi≥sgj
. Policy
π
schedules job
j
immediately following job
i
, while policy
π∗
is the same as
π
except for interchanging
jobs
i
and
j
. Under these settings, the following two theorems, which are obtained by using
Theorems 4and 5, can be used to compare these two totals.
Theorem 8. Suppose
CFXi(xi),FXj(xj)
is the joint survival function of
(Xi
,
Xj)
, where
C
is
the Archimedean copula with generator ϕsuch that ϕ−1is a completely monotone function. If
ϕ′(FXj)
ϕ′(FXi)
fXj
fXi
exp{ϕFXi−ϕFXj}
is increasing in t ∈R+, giis increasing for all i, and gi≥sgjfor all i and j, then
TCπ≤st TCπ∗.
Mathematics 2024,12, 3714 10 of 11
Theorem 9. Suppose
CFXi(xi),FXj(xj)
is the joint survival function of
(Xi
,
Xj)
, where
C
is an Archimedean copula with generator
ϕ
such that
ϕ−1
is a completely monotone function.
If
ϕFXj(t)−ϕFXi(t)
is decreasing in
t∈R+
,
gi
is increasing for all
i
, and
gi≥sgj
for all
i
and j, then
E(TCπ)≤E(TCπ∗).
6. Concluding Remarks
Suppose that
U
and
V
are two random variables. Furthermore, for all
α>
0, let
U(α)
and
V(α)
be two independent random variables such that the survival (distribution)
functions of
U(α)
and
V(α)
are
exp−αϕFU
(
exp{−αϕ(FU)}
) and
exp−αϕFV
(exp{−αϕ(FV)})
, respectively. Then, for all
α>
0, we have obtained the equivalence
relationship between the (reversed) hazard rate order (likelihood ratio order) of
U(α)
and
V(α)
and the functions of
U
and
V
. These equivalence relationships play a key role in
deriving the results of this paper.
Using these equivalence relationships, two sufficient and necessary conditions for
functions of two dependent random variables to preserve usual stochastic order were
provided. We have proved two sufficient conditions for the functions of two dependent
random variables based on the usual stochastic order. And we have also shown three
sufficient conditions for the characterization of different bivariate functions composed by
Archimedean copula-linked dependent random variables. These results generalize some
existing results in the literature and bring conclusions closer to reality. Two applications in
scheduling problems are also provided to illustrate the main results.
Due to the importance of stochastic order comparisons for dependent random vari-
ables, it is worth considering that the copulas in each set of the random variables are
different. We shall study this problem and plan to report these results in future work.
Author Contributions: Conceptualization, Q.G.; Methodology, Q.G. and P.J.; Formal analysis, Q.G.;
Writing—original draft, Q.G. and P.J.; Writing—review and editing, Q.G., P.J. and G.L.; Supervision,
P.J. All authors have read and agreed to the published version of the manuscript.
Funding: This work was supported by the Fujian natural science foundation (Grant No. 2024J01927),
Wuyi University Started the Project of Introducing Talents for Scientific Research (Grant No. YJ202316),
the Technical Consulting Project (Research on corporate financing scheme and business development
risk, Contract No. HX-2024-06-066), the Talent Cultivation and Research Start-up Foundation of
Anhui Polytechnic University (Grant No. S022022014).
Data Availability Statement: No new data were created or analyzed in this study. Data sharing is
not applicable to this article.
Conflicts of Interest: The authors declare no conflicts of interest.
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