# Extreme Value Theory: Lognormal GEV parameters

Lognormal distribution belongs to the Gumbel maximum domain of attraction, where:

$F^{logN}(x; \mu,\sigma)=\Phi\left(\frac{\ln x - \mu}{\sigma}\right)$,

$F^{Gum}(x;\mu,\beta) = e^{-\exp\left({-\frac{x-\mu}{\beta}}\right)}$

My question: Do we have $\mu=\mu$ and $\sigma=\beta$ ?

The Generalized Extreme Value distribution also uses notation $\beta=\sigma$ (Gumbel is the limit case $\xi =0$), and comparing the CDFs for Standard-Lognormal and Standard-Gumbel would again imply the parameters coincide. But I am not sure about it, because Gumbel is a limiting case of Lognormal Maxima, so there might be some transformation of the parameters aswell.

Let $$X_i \sim_{\text{i.i.d}} \text{LNorm}(\mu,\,\sigma)$$ with the meaning that the r.v. $$\log X_i$$ is normal with mean $$\mu$$ and standard deviation $$\sigma$$. Considering $$M_n := \max_{1 \leqslant i \leqslant n} X_i$$, we know that there exist two sequences $$a_n >0$$ and $$b_n$$ such that
$$\tag{1} \frac{M_n - b_n}{a_n} \to \text{Gum}(0, 1)$$
where $$\text{Gum}(\nu,\,\beta)$$ denotes the Gumbel distribution with location $$\nu$$ and scale $$\beta$$. This means that $$F_{M_n}(a_n x + b_n) \to F_{\text{Gum}}(x;\,0,\,1)$$ for all $$x$$.
Quite obviously the two sequences $$a_n$$ and $$b_n$$ depend on $$\mu$$ and $$\sigma$$, so they could be denoted as $$a_n(\mu,\,\sigma)$$ and $$b_n(\mu,\,\sigma)$$. For instance if $$\mu$$ is replaced by $$\mu +1$$ then the distribution of $$X_i$$ is replaced by that of $$e X_i$$ and the distribution of $$M_n$$ is replaced by that of $$e M_n$$, implying that $$a_n$$ and $$b_n$$ have to be replaced by $$ea_n$$ and $$eb_n$$ to maintain the same limit. Similarly if we replace $$\mu$$ by $$0$$ with $$\sigma$$ unchanged, $$X_i$$ is to be replaced by $$e^{-\mu} X_i$$ and then $$a_n$$ and $$b_n$$ must be replaced by $$e^{-\mu} a_n$$ and $$e^{-\mu}b_n$$.
The question can be formulated as: if we use the sequences $$a_n(0, 1)$$ and $$b_n(0, \,1)$$ at the left-hand side of (1) - instead of the due $$a_n(\mu,\,\sigma)$$ and $$b_n(\mu,\,\sigma)$$ - do we get $$\text{Gum}(\mu,\,\sigma)$$ at the right-hand side? The answer is then no, because the parameters of the Gumbel are indeed location and scale parameters, while this is not true for the log-normal. The parameter $$\sigma$$ of the log-normal impacts the tail, as can be seen by the fact that the coefficient of variation increases with $$\sigma$$. While $$\text{LNorm}(\mu,\,\sigma)$$ always remains in the Gumbel domain of attraction, the sequences $$a_n$$ and $$b_n$$ must tend to $$\infty$$ more rapidly as $$\sigma$$ increases. It can be proved that we can in (1) use sequences $$a_n$$ and $$b_n$$ such that $$b_n(\mu, \sigma) = e^\mu \, b_n(0, 1)^\sigma, \qquad a_n(\mu, \sigma) = \sigma \,(2 \log n)^{-1/2} b_n(\mu,\,\sigma),$$ see Embrechts P., Klüppelberg C. and Mikosch T. table 3.4.4 pp 155-157. If we use sequences $$a_n$$ and $$b_n$$ with a wrong $$\sigma$$, we will not get a non-degenerate limit for the left-hand side of (1), because the growth rates of $$a_n$$ and $$b_n$$ are then unsuitable for the tail of $$X_i$$.