Is there an asymptotic or approximate expression for the expectation (maybe even the variance, but my main question is about the mean) of the n-th order statistic, i.e. the maximum, of n independent, identically distributed Gaussian random variables from $\mathcal{N}(\mu,\sigma)$ where n is large?
I have tried plotting the estimate of Blom from here, but for very large quantities ($>2^{20}$) this seems to become overoptimistic.
This question was seemingly answered here, however plotting these formulas they seem to be for a standard normal distribution, so for $\mathcal{N}(0,1)$.
I however have a (very large) sample from $\mathcal{N}(\mu,\sigma)$. How do the formulas of the max-central limit theorem change in this case? For the the mean $\mu$ the mean of the probably just shifts by the same quantity. What happens with the $\sigma$?