General Closed Form and dispersion parameter of the Expected Maximum of i.i.d Gumbel Variables

I would like to know the general closed form of the expected maximum of i.i.d Gumbel variables. I found this onlie: Expectation of the Maximum of iid Gumbel Variables

In the linked page, it shows the log-sum form for $$E[\max_{i}(v_i+\epsilon_i)]$$ when each $$\epsilon_i$$ are i.i.d and follows $$G(\mu,1)$$ where the scale parameter $$\sigma=1$$.

But, I could not find the general form of the expected maximum when the scale parameter is not equal to 1. Especially, I am interested in the closed log-sum form when $$\mu=0$$.

Since this is in the line of random utility models, I found this: $$E[\max_{i}(v_i+\epsilon_i)]=ln(\sum_{i}exp(\frac{v_i}{\sigma}))+\gamma$$ where $$\gamma$$ is Euler constant and $$\sigma$$ as a dispersion parameter ($$\mu$$=0, $$\sigma\neq 1$$).

Is there any paper or proof that shows this solution with a dispersion parameter?

• The scale parameter, if it is common to all the variables, is irrelevant because it merely establishes a unit of measurement. Could you therefore clarify what your specific situation is? The full distribution of the maximum is obtained at stats.stackexchange.com/questions/192424, btw.
– whuber
Commented May 26, 2022 at 20:23
• Thank you. But, the link is the same link I posted. I would like to conduct a Maximum likelihood estimation for my model where $v_i$ is a function of some exogenous variables. The reason why I need the closed form solution is that my model is 2-period. So, in the first period, it will be $\max_{i1} (v_{i1}+\epsilon_{i1} +E[\max_{i2} (v_{i2}+\epsilon_{i2})])$. From the closed form of the Emax, I can derive the likelihood. That's the reason why I am looking for the log-sum form. And, some ppt files online just show the form I wrote above with a dispersion parameter to be estimated by the MLE. Commented May 26, 2022 at 20:29
• What distinction are you making between a "scale parameter" and a "dispersion parameter"?
– whuber
Commented May 26, 2022 at 20:35
• $\epsilon_i$ is an idiosyncratic shock, so it is assumed to be 0 at mean, but the model does not make an assumption about the scale parameter (another parameter to be estimated). My terminology may be confusing here, but according to my understanding, the purpose of the $\sigma$ showing up in the log sum form is that it will show up in the likelihood, so it can be estimated by the MLE. Commented May 26, 2022 at 20:40
• $v_i$ is a linear function of some exogenous variables which describe individual characteristics. Commented May 26, 2022 at 20:57