Approximate Order Statistics for lognormal variables Are there any known formulas that approximate the expected value of the maximum of $N$ i.i.d. lognormal random variables?
I am looking for something similar to:
Approximate order statistics for normal random variables
But in general any useful info on the expected value will be appreciated.
 A: I do not know of any standard approximate formulas for this. So I tried the following standard technique.
Let $X_1,...,X_N$ be $N$ iid log-normal random variables. Let $Y=\max_i X_i$ be their maximum. Now we need to find $E[Y].$ Let $F_Y(y)$ be the cdf of $Y$ and $F_X(x)$ be the cdf of any $X_i$'s. Then,
\begin{eqnarray}
F_Y(y) &=& P(Y \leq y) = P(\max_i X_i \leq y) \\
&=& P(\cap_i \{X_i \leq y\}) \\
&=& \prod_i P(X_i \leq y) = F_X(y)^N
\end{eqnarray}
Since $X_i$'s are iid. Now note that $X_i$'s are non-negative random variables and so is $Y.$ For any non-negative RV we can express its expectation in terms if its cdf as follows. You can find its derivation here.
$$
E[Y] = \int_0^\infty (1-F_Y(y))dy = \int_0^\infty (1-F_X(y)^N)dy
$$
Now for any log-normal random variable $F_X(x) = \Phi\left(\frac{\ln x - \mu}{\sigma} \right),$ where $\Phi$ is the cdf of the standard normal distribution.
Therefore 
$$
E[Y] = \int_0^\infty \left(1-\Phi\left(\frac{\ln y - \mu}{\sigma} \right)^N\right)dy
$$
Till now we have an exact formula for $E[Y].$ All I can think of now is a numerical integration to approximately compute $E[Y].$ The function $\Phi$ is well approximated and there are library functions for it. For a given $\mu,\sigma,N$ we can approximately compute $E[Y]$ using standard numerical integration techniques.
