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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

  1. When you scale $Z_j$ with a constant $A_j$, the distribution becomes
    $$A_jZ_j∼F(z_j)=\exp(−A^\theta_jz^{−θ}_j).$$

  2. The probability that $i=\arg \max_j[A_jZ_j]$ is given by
    $$\pi_i=\frac{A^\theta_i}{\sum_jA^\theta_j}.$$

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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

$\begin{aligned} \operatorname{Pr}\left(i=\arg \max _j\left[A_j Z_j\right]\right) & = \int_0^{\infty} \Pi_{j \neq i} \operatorname{Pr}\left(A_j Z_j \leq A_i Z_i \right) d F(Z_i) \\ & = \int_0^{\infty} \Pi_{j \neq i} \exp \left(-A_j^\theta (A_iZ_i)^{-\theta}\right) F'(Z_i) dZ_i \\ & = \int_0^{\infty} \exp \left(\sum_{j \neq i} -A_j^\theta (A_iZ_i)^{-\theta}\right) \theta Z_i^{-\theta-1}\exp \left(-Z_i^{-\theta}\right) d Z_i \\ & = \int_0^{\infty} \exp \left(\sum_{j } -A_j^\theta (A_iZ_i)^{-\theta}\right) \theta Z_i^{-\theta-1} d Z_i \\ & = \frac{A_i^\theta}{\sum_{j } A_j^\theta} \left[ \exp \left(\sum_{j } -A_j^\theta (A_iZ_i)^{-\theta}\right) \right]_0^{\infty} \\ & = \frac{A_i^\theta}{\sum_{j } A_j^\theta} \end{aligned}$

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  • $\begingroup$ Thank you for your answer. It was mostly clear. I have two questions: First, do you have a reference for these derivations? Second, in the first line of part 2, can you please explain how you write the integral with respect to the measure? $\endgroup$
    – nate
    Commented Jun 3 at 20:19
  • $\begingroup$ Here is a reference: sciencedirect.com/science/article/abs/pii/… $\endgroup$ Commented Jun 4 at 9:28

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