Solved – Expectation of the Maximum of iid Gumbel Variables

I keep reading in economics journals about a particular result used in random utility models. One version of the result is: if $epsilon_i sim_{iid}, $ Gumbel($mu, 1), forall i$, then:

$$E[max_i(delta_i + epsilon_i)] = mu + gamma + lnleft(sum_i expleft{delta_i right} right), $$

where $gamma approx 0.52277$ is the Euler-Mascheroni constant. I've checked that this makes sense using R, and it does. The CDF for the Gumbel$(mu, 1)$ distribution is:

$$G(epsilon_i) = exp(-exp(-(epsilon_i – mu)))$$

I'm trying to find a proof of this and I've had no success. I've tried to prove it myself but I can't get past a particular step.

Can anyone point me to a proof of this? If not, maybe I can post my attempted proof up to where I get stuck.

I appreciate the work exhibited in your answer: thank you for that contribution. The purpose of this post is to provide a simpler demonstration. The value of simplicity is revelation: we can easily obtain the entire distribution of the maximum, not just its expectation.

Ignore $mu$ by absorbing it into the $delta_i$ and assuming the $epsilon_i$ all have a Gumbel$(0,1)$ distribution. (That is, replace each $epsilon_i$ by $epsilon_i-mu$ and change $delta_i$ to $delta_i+mu$.) This does not change the random variable

$$X = max_{i}(delta_i + epsilon_i) = max_i((delta_i+mu) + (epsilon_i-mu)).$$

The independence of the $epsilon_i$ implies for all real $x$ that $Pr(Xle x)$ is the product of the individual chances $Pr(delta_i+epsilon_ile x)$. Taking logs and applying basic properties of exponentials yields

$$eqalign{ log Pr(Xle x) &= logprod_{i}Pr(delta_i + epsilon_i le x) = sum_i logPr(epsilon_i le x – delta_i)\ &= -sum_ie^{delta_i}, e^{-x} = -expleft(-x + logsum_i e^{delta_i}right). }$$

This is the logarithm of the CDF of a Gumbel distribution with location parameter $lambda=logsum_i e^{delta_i}.$ That is,

$X$ has a Gumbel$left(logsum_i e^{delta_i}, 1right)$ distribution.

This is much more information than requested. The mean of such a distribution is $gamma+lambda,$ entailing

$$mathbb{E}[X] = gamma + logsum_i e^{delta_i},$$


Similar Posts:

Rate this post

Leave a Comment