- Source: Tail dependence
In probability theory, the tail dependence of a pair of random variables is a measure of their comovements in the tails of the distributions. The concept is used in extreme value theory. Random variables that appear to exhibit no correlation can show tail dependence in extreme deviations. For instance, it is a stylized fact of stock returns that they commonly exhibit tail dependence.
Definition
The lower tail dependence is defined as
λ
ℓ
=
lim
q
→
0
P
(
X
2
≤
F
2
−
1
(
q
)
∣
X
1
≤
F
1
−
1
(
q
)
)
.
{\displaystyle \lambda _{\ell }=\lim _{q\rightarrow 0}\operatorname {P} (X_{2}\leq F_{2}^{-1}(q)\mid X_{1}\leq F_{1}^{-1}(q)).}
where
F
−
1
(
q
)
=
i
n
f
{
x
∈
R
:
F
(
x
)
≥
q
}
{\displaystyle F^{-1}(q)={\rm {inf}}\{x\in \mathbb {R} :F(x)\geq q\}}
,
that is, the inverse of the cumulative probability distribution function for q.
The upper tail dependence is defined analogously as
λ
u
=
lim
q
→
1
P
(
X
2
>
F
2
−
1
(
q
)
∣
X
1
>
F
1
−
1
(
q
)
)
.
{\displaystyle \lambda _{u}=\lim _{q\rightarrow 1}\operatorname {P} (X_{2}>F_{2}^{-1}(q)\mid X_{1}>F_{1}^{-1}(q)).}
See also
Correlation
Dependence
References
Kata Kunci Pencarian:
- Tail dependence
- Dependency
- Correlation
- Long-range dependence
- Long-tail traffic
- Copula (statistics)
- Portfolio optimization
- Extreme value theory
- Financial correlation
- Polyadenylation
