Environmental Engineering Reference
In-Depth Information
less than or equal to the 100
t
-th percentile of
F
2
given that
X
1
is less than or equal to the
100
t
-th percentile of
F
1
as
t
approaches 0 (e.g., Nelsen 2006)
2
1
−
−
1
λ
L
=
lim
PX
[
≤
FtXFt
( )
≤
( )]
(2.17)
2
1
1
+
t
→
0
If λ
L
∈ (0,1],
X
1
and
X
2
are said to be asymptotically dependent at the lower tail; if λ
L
= 0,
X
1
and
X
2
are said to be asymptotically independent at the lower tail.
It is noted that although the tail dependence is associated with the bivariate distribution of
X
1
and
X
2
, the two parameters λ
U
and λ
L
are nonparametric and depend only on the copula
function of
X
1
and
X
2
. Specifically, λ
U
can be expressed in terms of a copula function as
2
1
−
−
1
λ
U
=
lim
PX
[
>
FtXFt
( )
>
( )]
2
1
1
−
t
→
1
PX
[
>
FtXF
−
1
( ),
>
−
1
()]
t
1
1
2
2
=
lim
PX
[
>
Ft
−
1
( )]
−
t
→
1
1
1
1
−
PX
[
≤
Ft
−
1
( )]
−
PX
[
≤
Ft
2
1
−
( )]
+
PX
[
≤
FtXFt
−
1
( ),
≤
2
1
−
()]
1
1
2
1
1
2
=
lim
1
−
PX
[
≤
Ft
−
1
( )]
−
t
→
1
1
1
12
−+
tCt
(,
t
;θ
=
lim
(2.18)
1 −
t
→
−
t
1
Similarly, λ
L
can be expressed in terms of a copula function as
λ
L
=
lim
PX
[
≤
FtXFt
2
1
−
( )
≤
−
1
( )]
2
1
1
+
t
→
0
PX
[
≤
FtXF
−
1
( ),
≤
2
1
−
()]
t
1
1
2
=
lim
PX
[
≤
Ft
−
1
( )]
+
t
→
0
1
1
Ct t
t
(, ;)
θ
=
lim
(2.19)
+
t
→
0
Equations 2.18
and
2.19
clearly state that the tail dependence of a bivariate distribu-
tion can be directly obtained from its copula function. As mentioned in the literature (e.g.,
Nelsen 2006), the Gaussian, Plackett, and Frank copulas do not have tail dependence. That
is, λ
U
= 0 and λ
L
= 0 for the Gaussian, Plackett, and Frank copulas. Unlike the aforemen-
tioned three copulas, the No.16 copula has only lower tail dependence and the coefficient
of lower tail dependence λ
L
is a constant of 0.5 for the whole range of the θ parameter. The
lower tail dependence underlying the No.16 copula can be observed in
Figure 2.2d
for a
relatively small correlation coefficient τ = −1/2. It is noted that although λ
L
is a constant,
the effect of this lower tail dependence is reduced and the No.16 copula becomes approxi-
mately radially symmetric when the negative correlation becomes strong (e.g., Nelsen 2006).
This phenomenon can be observed by comparing
Figure 2.2d
(τ = −1/2) with
Figure 2.2c
(τ = −1/3). In general, the lower tail dependence of the No.16 copula has an important effect
on reliability (see Section 2.5), which should be noted in practical applications.
It is known that a key step in copula modeling is the determination of copula param-
eters. In the previous applications, the copula parameters θ underlying the four copulas are
obtained from the Kendall rank correlation coefficients τ using
Equation 2.13
. This is a dual
integral equation. Solving this equation needs great efforts. In the following equations, some
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