Tail dependence

The lower tail dependence is defined as^[2]

${\displaystyle \lambda _{\ell }=\lim _{q\rightarrow 0}\operatorname {P} (X_{2}\leq F_{2}^{-1}(q)\mid X_{1}\leq F_{1}^{-1}(q)).}$

where${\displaystyle F^{-1}(q)={\rm {inf}}\{x\in \mathbb {R} :F(x)\geq q\}}$ ,that is, the inverse of the cumulative probability distribution function for q.

The upper tail dependence is defined analogously as

${\displaystyle \lambda _{u}=\lim _{q\rightarrow 1}\operatorname {P} (X_{2}>F_{2}^{-1}(q)\mid X_{1}>F_{1}^{-1}(q)).}$

• Correlation
• Dependence