Properties of Multivariate Frechet Distributions

External question — Economics Stack Exchange Author: nate Original post: https://economics.stackexchange.com/questions/58547 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. I was reading an answer from another Economics Stack Exchange post and I had further questions. The highest up-voted answer here utilized two properties of multivariate Frechet distributions, but the author did not prove them. I am wondering if anyone (or the author of the answer) can provide a reference where I can learn more about these properties. For convenience, the I provide the properties below: If $Z_j$ are independent random variables with Frechet CDFs $F(z_j) = \exp(-z_j ^{- \theta})$ , then When you scale $Z_j$ with a constant $A_j$ , the distribution becomes $$A_jZ_j∼F(z_j)=\exp(−A^\theta_jz^{−θ}_j).$$ The probability that $i=\arg \max_j[A_jZ_j]$ is given by $$\pi_i=\frac{A^\theta_i}{\sum_jA^\theta_j}.$$

External answer — Economics Stack Exchange Author: Alalalalaki Original post: https://economics.stackexchange.com/a/58549 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. 1 $\begin{aligned} \operatorname{Pr}\left(A_j Z_j \leq z\right)=\operatorname{Pr}\left(Z_j \leq \frac{z}{A_j}\right)=F\left(\frac{z}{A_j}\right) =\exp \left(-\left(\frac{z}{A_j}\right)^{-\theta}\right)=\exp \left(-A_j^\theta z^{-\theta}\right) \end{aligned}$ 2 $\begin{aligned} \operatorname{Pr}\left(i=\arg \max _j\left[A_j Z_j\right]\right) & = \int_0^{\infty} \Pi_{j \neq i} \operatorname{Pr}\left(A_j Z_j \leq A_i Z_i \right) d F(Z_i) \\ & = \int_0^{\infty} \Pi_{j \neq i} \exp \left(-A_j^\theta (A_iZ_i)^{-\theta}\right) F'(Z_i) dZ_i \\ & = \int_0^{\infty} \exp \left(\sum_{j \neq i} -A_j^\theta (A_iZ_i)^{-\theta}\right) \theta Z_i^{-\theta-1}\exp \left(-Z_i^{-\theta}\right) d Z_i \\ & = \int_0^{\infty} \exp \left(\sum_{j } -A_j^\theta (A_iZ_i)^{-\theta}\right) \theta Z_i^{-\theta-1} d Z_i \\ & = \frac{A_i^\theta}{\sum_{j } A_j^\theta} \left[ \exp \left(\sum_{j } -A_j^\theta (A_iZ_i)^{-\theta}\right) \right]_0^{\infty} \\ & = \frac{A_i^\theta}{\sum_{j } A_j^\theta} \end{aligned}$