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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?

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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 $$ \alpha_r = \alpha_{r-1} + \binom{N-1}{r-1}J(N-r+1,r-1)$$

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    $\begingroup$ Thanks! I also found the following slightly more explicit formula for the expectation of the maximum on a book by Chin-Diew Lai called "Generalized Weibull Distributions" $$ E[max(X_1,...,X_n)] = n \Gamma(1+1/\beta) \sum_{i=0}^{n-1} \binom{n-1}{i} (-1)^i (\frac{1}{i+1})^{1 + 1/\beta}$$ $\endgroup$ – Asterix Jun 20 '17 at 17:22

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