<p>1</p>
<p><span class="math-container">$\begin{aligned} \operatorname{Pr}\left(A_j Z_j \leq z\right)=\operatorname{Pr}\left(Z_j \leq \frac{z}{A_j}\right)=F\left(\frac{z}{A_j}\right) =\exp \left(-\left(\frac{z}{A_j}\right)^{-\theta}\right)=\exp \left(-A_j^\theta z^{-\theta}\right)  \end{aligned}$</span></p>
<p>2</p>
<p><span class="math-container">$\begin{aligned} \operatorname{Pr}\left(i=\arg \max _j\left[A_j Z_j\right]\right) &amp; = \int_0^{\infty} \Pi_{j \neq i} \operatorname{Pr}\left(A_j Z_j \leq A_i Z_i  \right) d F(Z_i) \\ &amp; =  \int_0^{\infty} \Pi_{j \neq i} \exp \left(-A_j^\theta (A_iZ_i)^{-\theta}\right) F'(Z_i) dZ_i  \\ &amp; =  \int_0^{\infty}  \exp \left(\sum_{j \neq i} -A_j^\theta (A_iZ_i)^{-\theta}\right) \theta Z_i^{-\theta-1}\exp \left(-Z_i^{-\theta}\right) d Z_i  \\ &amp; =  \int_0^{\infty}  \exp \left(\sum_{j } -A_j^\theta (A_iZ_i)^{-\theta}\right) \theta Z_i^{-\theta-1} d Z_i  \\ &amp; = \frac{A_i^\theta}{\sum_{j } A_j^\theta} \left[ \exp \left(\sum_{j } -A_j^\theta (A_iZ_i)^{-\theta}\right)  \right]_0^{\infty} \\ &amp; = \frac{A_i^\theta}{\sum_{j } A_j^\theta}  \end{aligned}$</span></p>

CC BY-SA 4.0
2024-06-03T20:19:27.097Z
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?
nate
CC BY-SA 4.0
2024-06-04T09:28:39.140Z
Here is a reference: https://www.sciencedirect.com/science/article/abs/pii/S0191261513001987?via%3Dihub
Jesper Hybel
CC BY-SA 4.0
2024-06-03T17:48:34.360Z
2024-06-03T17:48:34.360Z
Alalalalaki
<p>I was reading an answer from another Economics Stack Exchange post and I had further questions.</p>
<p>The highest up-voted answer <a href="https://economics.stackexchange.com/questions/44233/optimal-residential-location-and-multivariate-frechet-distribution">here</a> 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:</p>
<p>If <span class="math-container">$Z_j$</span> are independent random variables with Frechet CDFs <span class="math-container">$F(z_j) = \exp(-z_j ^{- \theta})$</span>, then</p>
<ol>
<li><p>When you scale <span class="math-container">$Z_j$</span> with a constant <span class="math-container">$A_j$</span>, the distribution becomes<br />
<span class="math-container">$$A_jZ_j∼F(z_j)=\exp(−A^\theta_jz^{−θ}_j).$$</span></p>
</li>
<li><p>The probability that <span class="math-container">$i=\arg \max_j[A_jZ_j]$</span> is given by<br />
<span class="math-container">$$\pi_i=\frac{A^\theta_i}{\sum_jA^\theta_j}.$$</span></p>
</li>
</ol>

<microeconomics><macroeconomics><mathematical-economics><statistics>
Properties of Multivariate Frechet Distributions
nate
# Properties of Multivariate Frechet Distributions

(Asked by: username_0 on 2024-06-03)

I was reading an answer from another Economics Stack Exchange post and I had further questions.

The highest up-voted answer [here](https://economics.stackexchange.com/questions/44233/optimal-residential-location-and-multivariate-frechet-distribution) 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  
   $$\pi\_i=\frac{A^\theta\_i}{\sum\_jA^\theta\_j}.$$

**Tags:** microeconomics, macroeconomics, mathematical-economics, statistics


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## 1 answers

### Answer by username_1 on 2024-06-03. Score: 3 | Accepted answer

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

$\begin{aligned} \operatorname{Pr}\left(i=\arg \max \_j\left[A\_j Z\_j\right]\right) & = \int\_0^{\infty} \Pi\_{j \neq i} \operatorname{Pr}\left(A\_j Z\_j \leq A\_i Z\_i \right) d F(Z\_i) \\ & = \int\_0^{\infty} \Pi\_{j \neq i} \exp \left(-A\_j^\theta (A\_iZ\_i)^{-\theta}\right) F'(Z\_i) dZ\_i \\ & = \int\_0^{\infty} \exp \left(\sum\_{j \neq i} -A\_j^\theta (A\_iZ\_i)^{-\theta}\right) \theta Z\_i^{-\theta-1}\exp \left(-Z\_i^{-\theta}\right) d Z\_i \\ & = \int\_0^{\infty} \exp \left(\sum\_{j } -A\_j^\theta (A\_iZ\_i)^{-\theta}\right) \theta Z\_i^{-\theta-1} d Z\_i \\ & = \frac{A\_i^\theta}{\sum\_{j } A\_j^\theta} \left[ \exp \left(\sum\_{j } -A\_j^\theta (A\_iZ\_i)^{-\theta}\right) \right]\_0^{\infty} \\ & = \frac{A\_i^\theta}{\sum\_{j } A\_j^\theta} \end{aligned}$

> **Comments:**
> **username_0**: 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?  
> **username_2**: Here is a reference: https://www.sciencedirect.com/science/article/abs/pii/S0191261513001987?via%3Dihub  

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