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Consider $M$ i.i.d. random variables $V_1,..., V_M$ distributed as Gumbel with location $\lambda$ and scale $\beta$. We know that (see proof at the end of the question) $$ E(\max_{k\in \mathcal{Y}} V_k+\alpha_{k})= \log (\sum_{k=1}^M\exp(\frac{\alpha_{k}+\lambda}{\beta}) )\beta +\beta\gamma $$ where $E$ denotes expectation, $\mathcal{Y}\equiv \{1,...,M\}$, and $\gamma$ is the Euler constant, for any $a\equiv (a_1,..., a_M)\in \mathbb{R}^M$.

I have a doubt on a source that I am reading which claims that "[...] we proceed by centering all Gumbel distributions; the only effect of this normalisation is that it eliminates the Euler constant from the expectation."

I am confused how I should exactly interpret "centering". What is the new mean and new scale and why the Euler constant should disappear from the expression above?


Proof of expression above: for any $r\in \mathbb{R}$

$$ \begin{aligned} & \log(\mathbb{P}(\max_{k\in \mathcal{Y}} V_k+\alpha_{k}\leq r))= \log(\mathbb{P}(V_k+\alpha_{k}\leq r \text{ }\forall k \in \mathcal{Y}))=\log(\Pi_{k=1}^M\mathbb{P}(V_k+\alpha_{k}\leq r))\\ &=\sum_{k=1}^M \log(\mathbb{P}(V_k\leq r-\alpha_{k}))=\sum_{k=1}^M \log(\exp(-\exp(\frac{-r+\alpha_{k}+\lambda}{\beta})))\\ &= - \sum_{k=1}^M \exp(\frac{-r+\alpha_{k}+\lambda}{\beta})= - \sum_{k=1}^M \exp(\frac{-r}{\beta})\exp(\frac{\alpha_{k}+\lambda}{\beta})=- \exp(\frac{-r}{\beta})\sum_{k=1}^M\exp(\frac{\alpha_{k}+\lambda}{\beta})\\ & = - \exp \Big[ \log \Big( \exp(\frac{-r}{\beta})\sum_{k=1}^M\exp(\frac{\alpha_{k}+\lambda}{\beta}) \Big) \Big]= - \exp \Big[ \frac{-r}{\beta}+ \log \Big(\sum_{k=1}^M\exp(\frac{\alpha_{k}+\lambda}{\beta}) \Big) \Big]\\ &= \log\Big[ \exp \Big( - \exp (\frac{-r}{\beta}+ \log (\sum_{k=1}^M\exp(\frac{\alpha_{k}+\lambda}{\beta}) ))\Big) \Big]\\ &= \log\Big[ \exp \Big( - \exp (\frac{-r}{\beta}+\frac{ \log (\sum_{k=1}^M\exp(\frac{\alpha_{k}+\lambda}{\beta}) )\beta }{\beta})\Big) \Big] \end{aligned} $$ which is the log of the cdf of a Gumbel with scale $\beta$ and location $\log (\sum_{k=1}^M\exp(\frac{\alpha_{k}+\lambda}{\beta}) )\beta $. Hence, $$ \mathbb{E}_{\mathbb{P}}(\max_{k\in \mathcal{Y}} V_k+\alpha_{k})= \log (\sum_{k=1}^M\exp(\frac{\alpha_{k}+\lambda}{\beta}) )\beta +\beta\gamma $$

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It is common for a Gumbel distribution to have a location parameter $\mu$ and a scale parameter $\beta$

where $\mu$ is the mode and $\beta$ is $\frac{\sqrt{6}}{\pi}$ times the standard deviation

and then the mean is $\mu +\beta \gamma$ with $\gamma\approx 0.5772156649$ being the Euler–Mascheroni constant

so I suppose if you recentre by subtracting $\beta\gamma$ from all values then the new mean would be $\mu$ and you would have removed the Euler–Mascheroni constant from the mean, though as a side-effect would have introduced it to the mode

Your original expression might then be something like

Consider $M$ i.i.d. random variables $V_1,..., V_M$ distributed as Gumbel with location $\lambda - \beta \gamma$ and scale $\beta$. We know that $$ E\left(\max_{k\in \mathcal{Y}} V_k+\alpha_{k}\right)= \log \left(\sum_{k=1}^M\exp\left(\frac{\alpha_{k}+\lambda}{\beta}\right) \right)\beta $$

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