Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I have been given this formula for upper tail dependence and read that tail dependence depends on the copula and not the marginals: $$ \lambda_U = \lim_{a \to 1} \Pr[Y>F_Y^{-1}(a)\mid X>F_X^{-1}(a)] . $$

Would the inverse functions $F_X^{-1}(a)$ and $F_Y^{-1}(a)$ be copula functions ?

I have an intuitive understanding of what couplas are, but my knowledge of stats is quite weak, I would appreciate if you provide simplistic answers where possible.

share|improve this question
The limit as $a \to 1$ is missing in your formula for $\lambda_U$. – QuantIbex Apr 22 '14 at 12:45
Also, the conditionning should be on $X > F_x^{-1}(a)$, not $Y > F_x^{-1}(a)$ . – QuantIbex Apr 22 '14 at 12:47
This question provides the link between the upper tail dependence coefficient and the copula of $X$and $Y$. – QuantIbex Apr 22 '14 at 12:54
Hi @QuantIbex from the post you linked to above, I see that the functions in the formula for $lambda_u$ are not the coupla, but I don't understand from that post where the coupla fits in – AfterWorkGuinness Apr 22 '14 at 14:15
The link I provided gives the expression $$ \lambda_U = \lim_{u \to 1} \frac{1 - 2u + C(u,u)}{1-u}, $$ where $C$ is the copula (not "coupla") associated with $X$ and $Y$. Obviously, $\lambda_U$ depends on the copula, but not on the margins. @ocram's answer shows a way to derive this expression from the definition of $\lambda_U$. Please clarify if you need more information. – QuantIbex Apr 22 '14 at 15:21
up vote 6 down vote accepted

\begin{align} \Pr & \big[ Y > F_Y^{-1}(a) \,\big| X > F_X^{-1}(a) \big] \\[1em] & = \frac{\Pr \big[ Y > F_Y^{-1}(a), \, X > F_X^{-1}(a) \big]}{\Pr \big[ X > F_X^{-1}(a) \big]} \\[1em] & = \frac{\Pr \big[ Y < F_Y^{-1}(a), \, X < F_X^{-1}(a) \big] + \Pr \big[ X > F_X^{-1}(a) \big] + \Pr \big[ Y > F_Y^{-1}(a) \big] - 1}{1 - \Pr \big[ X < F_X^{-1}(a) \big]} \\[1em] & = \frac{C\big(a, a\big) + 1 - 2a}{1 - a} \end{align}

Thus, ''tail dependence [$\lambda_U$] depends on the copula and not the marginals.´´


$$ \Pr \big[ A \,\big| B \big] = \frac{\Pr[A,\, B]}{\Pr[B]} $$

$$ \Pr[X > x] = 1 - \Pr[X < x] $$

$$ \Pr[X > x, Y > y] = \Pr[X < x, Y < y] + \Pr[X > x] + \Pr[Y > y] - 1 $$

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.