From Wikipedia, the free encyclopedia
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.
[1]
Definition
The lower tail dependence is defined as
[2]
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
References
^ Hartmann, Philip; Straetmans, Stefan T.M.; De Vries, Casper G. (2004).
"Asset Market Linkages in Crisis Periods" . Review of Economics and Statistics . 86 (1): 313–326.
doi :
10.1162/003465304323023831 .
hdl :
10419/152505 .
S2CID
56001186 .
^ McNeil, Alexander J.; Frey, Rüdiger; Embrechts, Paul (2005),
Quantitative Risk Management. Concepts, Techniques and Tools , Princeton Series in Finance, Princeton, NJ: Princeton University Press,
ISBN
978-0-691-12255-7 ,
MR
2175089 ,
Zbl
1089.91037