0
$\begingroup$

In an example exercise they question whether it is possible to construct a bivariate distribution of $LN(0,1)$- and $LN(0,4)$-distributed random variables, where $LN(\mu,\sigma^2)$ is the log normal distribution, such that the linear correlation coefficient $\rho$ is -0.2.

In the answer they say that it can be proven that the minimum corr. coeff. $\rho_{min}$ is given when the two variables are countermonotonic. Thus, $\rho_{min}$ can be found by taking their countermonotonic representation and thereafter it is just straight forward calculations. They find that the minimum correlation coefficient is approximately equal to -0.09. Hence it is not possible to construct such bivariate distribution.

What I wonder is how can I find the countermonotonic representation?

The countermonotonic representation of the log normal r.v.'s is given by $(e^{-Z},e^{2Z})$, where Z is standard normal. But they do not explicitly explain why that is.

$\endgroup$
0
$\begingroup$

I think you can represent the countermonotonic of $\log$ normal by $(1-F(x), F(y))$; $1-F(x)$ implies after the calculations $e^{-Z}$ where $Z$ is $N(0,1)$ distributed.

Have a look on this site (in pdf form).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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