# Finding the mean of the max order statistic drawn from standard normal

Let $$X=\max\{X_1, X_2, \cdots, X_N\}$$, where each $$X_i \sim N(0,1)$$ and are independent. What is the approximate value of $$X$$ for large $$N$$.

The term "approximate" isn't defined very clearly. I'm assuming it means the mean, but I can't really figure out a nice way to calculate that, since the PDF of the $$N$$th order statistic uses CDF of the normal.

Any thoughts would be appreciated.

The distribution (asymptotic and exact) of the maximum order statistic from an iid normal sample is discussed here: Determine the limiting distribution of Standard Normal order statistics. Then the expectation can be found by numerical integration (example given there), or the asymptotic solution given there, which is a Gumbel distribution, can be used. That wikipedia page gives the expectation of a standard Gumbel variate as the Euler-Mascheroni constant $$\gamma\approx 0.57721$$, so with notation from the first linked page, the approximate expectation of the maximum of $$n$$ iid $$\mathcal{N}(0,1)$$ variables is $$a_{(n)} + b_{(n)} \gamma$$.

This question was posed in a different form some years ago on math Stack Exchange, where it was asked to prove that the maximum of $$n$$ independent standard Normals was asymptotically equivalent to $$\sqrt{2 \log n}$$.

After deriving the exact distribution of the maximum order statistic, it became immediately apparent that for large $$n$$ (and by that I do not mean just $$n = 100$$, but say $$n = 1,000,000$$ or even a billion), the proposed asymptote was appallingly useless, and moreover that there is no meaningful approximate value for $$X$$. The value of $$X$$ does not converge to any constant for large $$n$$.

The full posting on math SE is at:

https://math.stackexchange.com/questions/961780/prove-that-the-maximum-of-n-independent-standard-normal-random-variables-is-a

In brief, in answer to your question: there is no meaningful approximate value of $$X$$ for large $$n$$.

I think you want to look at the Fisher-Tippet Theorem which provides a CLT for the maximum, which states that the normalised maximum converges almost surely to one of three distributions (Gumbel, Frechet or Weibull)

• A key aspect of the Fisher-Tippet-Gnedenko Theorem is that the maximum has to be suitably normalized for the limit to exist. In particular, as $N\to\infty$, the maximum of $N$ iid Normal distributions diverges: it does not approach any particular distribution.
– whuber
Commented Jan 23, 2018 at 18:31