distribution for scaled Maximum of n independent Weibulls for $n \to \infty$ Assume that $X_1, X_2,...\sim Weibull(\lambda, k) \quad iid.$, i.e. 
$F(X_1\leq x) = 1-e^{-(\lambda x)^k}$
define $M_n:= \max\{X_1, ..., X_n\}$ and $\tilde{M}_n:=\frac{M_n-b_n}{a_n}$
according to extreme value theory the distribution of $\tilde{M}_n$ approaches for a proper choice of $a_n,b_n$ a distribution function: 
$$G(x)= \exp\left\{-\left[1+\xi\left(\frac{x-\mu}{\sigma}\right)\right]^{-1/\xi}\right\}$$. 
where $\mu \in \mathbb{R}, \sigma>0, \xi \in \mathbb{R}$ the case $\xi=0$ is interpreted as $\xi \to 0$
I.e.: $$\left\{1-\exp\left[-\lambda^k \left(\frac{x-b_n}{a_n}\right)^k\right]\right\}^n\to G(z), \quad n \to \infty$$
the expression on the left side can be reformulated as : 
$$\left\{1-\exp\left[-\lambda^k \left(\frac{x-b_n}{a_n}\right)^k\right]\right\}^n=\left\{1- \frac{n\exp\left[-\lambda^k \left(\frac{x-b_n}{a_n}\right)^k\right]}{n}\right\}^n$$
so I need to find sequences $a_n,b_n$, such that 
$$n\exp\left[-\lambda^k \left(\frac{x-b_n}{a_n}\right)^k\right]\to \left[1+\xi\left(\frac{x-\mu}{\sigma}\right)\right]^{-1/\xi}, n \to \infty$$
I would be glad if you could help me finding these sequences.
 A: The determination of the domain of attraction and of the related
constants $a_n$ and $b_n$ uses several functions related to the survival
function, here given by $S(x) = \exp\{-(\lambda x)^k\}$ for $x > 0$.
Of major importance are the tail-quantile function $U(t)$ and the
hazard rate function $h(x)$.
The tail-quantile is obtained by solving $S(U) = 1/t$ for $U$, which
leads to $U(t) = \lambda^{-1} (\log t)^{1/k}$ for $t> 1$. The hazard
rate is $h(x) = -\text{d}\log S(x)/\text{d}x = \lambda^k \, k
\,x^{k-1}$ for $x>0$.  The derivative of the inverse hazard rate
$1/h(x)$ is proportional to $x^{-k}$, hence tends to zero for large
$x$ since $k>0$. So the Von Mises' condition holds.  We know that
the distribution is in the Gumbel domain of attraction and that we
have the following convergence in distribution to the standard Gumbel
\begin{equation}
\frac{M_n - U(n)}{a_n} \to \text{Gumbel}.
\end{equation}
Moreover we know that we can choose $a_n$ as $e(U(n))$ where $e(x)$
denotes the mean residual life function given by $e(x) := A(x) /
S(x)$ with $A(x) := \int_{x}^\infty S(t) \,\text{d}t$ for $x >0$.
We now proceed to the evaluation of $e(U(n))$ or to the
determination of a quantity which is equivalent to it for large
$n$. Using the change of variable $u := (\lambda t)^k$ we get
$$
  A(x) = \int_{x}^\infty \exp\{-(\lambda t)^k\}\,\text{d}t= \frac{1}{\lambda k}
  \int_{(\lambda x)^k}^{\infty} u^{1/k -1} e^{-u} \text{d}u 
    = (\lambda k)^{-1} \, \Gamma(s,\, v),
$$
where $\Gamma(s,\,v)$ stands for the incomplete gamma
function
evaluated at $s:= 1/k$ and $v := (\lambda x)^k$. We can use the
following known result about the incomplete gamma $\Gamma(s,\, v) \sim
v^{s-1} e^{-v}$ for $v \to \infty$ which can be shown using
integration by parts. So
$$
e(x) \sim 
\frac{(\lambda k)^{-1}\, \left\{ (\lambda x)^k  \right\}^{1/k - 1}
\exp\{- (\lambda x)^k \}}{S(x)} = \frac{x}{k} \, (\lambda x)^{-k}.
$$
Note that $h(x) \times e(x)$ tends to $1$ for large $x$, which is
clear from the last equivalence; this limit condition is both
necessary an sufficient for the attraction to the Gumbel when $h(x)$
is monotonic for large $x$, as is the case here - see Theorem 1 in
Galambos and
Obretenov.
We can choose $a_n$ as $1 / h(U(n))$, and our constants can
be
$$
    a_n = \dfrac{1}{\lambda k} \, (\log n)^{1/k -1}, \qquad
    b_n = \dfrac{1}{\lambda} \, (\log n )^{1/k}.
$$
A precise statement of Von Mises' condition is found in the classical
book Modelling Extremal Events by Embrechts
P., Klüppelberg C. and Mikosch T. In this book (up to change in
notations), the couple of constants is given in Table 3.4.4.
## Weibull parameters
k <- 2.5; lambda <- 10
## simulate
set.seed(123)
n <- 40; nsim <- 10000
X <- array(rweibull(n * nsim, shape = k, scale = 1/ lambda), dim = c(nsim, n))
M <- apply(X, 1, max)
bn <- log(n)^(1 / k) / lambda
an <- log(n)^(1 / k - 1) / lambda / k
Mscale <- scale(M, center = bn, scale = an)
hist(Mscale, breaks = 100, probability = TRUE, col = "lightyellow",
     main = sprintf("Simulated maxima for n = %d", n), xlab = "")
require(evd)
curve(dgumbel, add = TRUE, col = "orangered", lwd  = 2)


