I'm trying to find an asymptotic approximation for the expectation of the maximum of $n$ Weibull random variables $X_i \sim Weibull(\lambda,\beta)$ when $\beta < 1/2$ and $n$ is large. From simulations, I'm getting that
$$E[max(X_1,...,X_n) ] \approx A n^{c}$$ for some constant $A$ and constant $c < 1$, but I'm not sure how to prove something like this.
Are there any well-known approximations for the expectation of the maximum of Weibull random variables?
An exact formula is given by "A note on order statistics from Weibull distribution" by Balakrishnan and Joshi. Define $$J(p,0) = \frac{\Gamma\left(\frac{1}{\beta}\right)}{\lambda\beta p^{1/\beta}}$$ with $$J(p,m) = J(p,m-1) - J(p+1,m-1) $$ Define $\alpha_n$ as the expected value of the $n$th smallest sample. Then the expected value of the smallest sample is $$\alpha_1 = J(1,0)$$ and for larger samples is $$ \alpha_r = \alpha_{r-1} + \binom{N-1}{r-1}J(N-r+1,r-1)$$
$$ E[max(X_1,...,X_n)] = n \Gamma(1+1/\beta) \sum_{i=0}^{n-1} \binom{n-1}{i} (-1)^i (\frac{1}{i+1})^{1 + 1/\beta}$$
– Asterix Jun 20 '17 at 17:22