# Expected value of a Gumbel variable conditional on Gumbel being the maximum of N iid Gumbel

I found the following results in Hanemann (1984) which I cannot find a proof for. I checked through simulation that it is right, but I would like to see an analytical proof...

Hanemann, W. M. (1984). Discrete/continuous models of consumer demand. Econometrica: Journal of the Econometric Society, 541-561.

He considers a discrete choice with $$i=1,...,N$$ options. The value of a choice is the sum of an intercept that is specific to each option $$v_i$$ and a random draw: $$V_{i}^{F} = v_i + \zeta_{i}$$ where $$\zeta_{i}$$ is an idiosyncratic taste shock.

He defines with $$M_i$$ for the set of values of the vector $$\zeta$$ such that the option $$i$$ yields to highest to the agent, i.e. $$A_i \equiv \left\{\zeta \mid V_{i}^{F}\geq V_{j}^{F}, \; \forall j \right\}$$.

He further considers the case where $$\zeta$$ is a vector of i.i.d. random variables distributed Type 1 Extreme Value with scale/dispersion parameter $$\sigma$$. Note that $${P}(\zeta \in A_{i})$$ is the probability that option $$i$$ is actually chosen.

Hanemann (1984) states the following results in equation (3.15) and I was wondering if one knew where to find the proof. $$$${E}\left\{e^{t \zeta_{i}} \mid \zeta \in A_{i}\right\} = \Gamma(1-\sigma t) \times \beta_{i}^{\sigma t}; \quad \text{ where } \beta^{-1}_{i} = {P}(\zeta \in A_{i}).$$$$

• Could you rewrite the question without using economic terms? Jan 23, 2021 at 14:58

Let $$\boldsymbol{\zeta} = [\zeta_i]_{1 \leq i \leq N}$$ be a random vector with i.i.d. components $$\zeta_i$$ being Gumbel with location $$\mu = 0$$ and scale $$\sigma$$. Let $$B_i = \{\ \boldsymbol{\zeta} \in A_i\}$$ be the event "$$\zeta_i$$ is the largest of the $$N$$ r.vs. $$\zeta_j$$". Then the claim is that $$\mathbb{E}\{ e^{t\zeta_i} \vert B_i \} = \Gamma(1 - \sigma t) \times \beta_i^{\sigma t}$$ where $$\beta_i = 1 / \text{Pr}(B_i)$$.
Note that since the r.vs $$\zeta_i$$ are i.i.d we have $$\beta_i = N$$ because each of the $$N$$ r.vs $$\zeta_i$$ has the same probability of being the largest. With $$M := \max_i \{\zeta_i\}$$ we have for any $$t$$ $$\mathbb{E}\{ e^{t\zeta_i} \vert B_i \} = \mathbb{E}\{ e^{t M} \vert B_i \} = \mathbb{E}\{ e^{t M} \}$$ because the distribution of $$M$$ conditional on $$B_i$$ does not depend on the specific index $$i$$ where the maximum is reached.
Moreover it easy to see that $$M$$ is Gumbel with location $$\mu = \sigma \log N$$ and scale $$\sigma$$. Using the moment generating function of the Gumbel distribution with location $$\mu$$ and scale $$\sigma$$ as given in Wikipedia, that is $$\mathbb{E}\{ e^{tM} \} = \Gamma(1 - \sigma t) \times e^{\mu t},$$ we get the wanted result because $$e^{\mu t} = e^{\sigma t \log N} = N^{\sigma t}$$.