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How does the maximum of $N$ normal random variables behave? If they are i.i.d., then on the right tail ($x \rightarrow \infty$) it behaves as if it is distributed with Gumbel. But any results for the generic case?

That is, how does $Z = \max \left(X_1, X_2, \dots, X_N \right)$ behave where $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$, or with shared variance $X_i \sim \mathcal{N}(\mu_i, \sigma^2)$.

I assume the shared variance & different mean case should behave similar to the i.i.d. case, but are there any known results?

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This problem is an example of order statistics with non-identical distributions; in this instance, due to having non-identical parameters. We are given a $N(\mu, \sigma^2)$ parent where identicality is relaxed by replacing $(\mu, \sigma)$ with $(\mu_i, \sigma_i)$ for $i = 1, ..., n$, with parent pdf say $f_i(x)$:

enter image description here

and cdf $F_i(x)$.

The general form for the pdf of $Z = \max \left(X_1, X_2, \dots, X_n \right)$ will be:

$$\text{pdf}(Z) \quad = \quad \sum _{i=1}^n \big\{ \; f_i(z) \prod _{j=1}^n F_j(z){}_{\text{for } j\neq i} \big \}$$

This is manageable. To illustrate and check, consider a specific case of say $n = 4$. Then the pdf of the maximum is the pdf of the $4^{\text{th}}$ order statistic in a sample of size 4, which can be evaluated automatically using the OrderStatNonIdentical function in the mathStatica package for Mathematica:

enter image description here

where the Normal cdf $F_i(x) = \frac{1}{2} \left(1+ \text{Erf}\left(\frac{x-\mu _i}{\sqrt{2} \sigma _i}\right)\right)$, and Erf denotes the error function.

This can yield unusual functional forms, such as bimodal pdfs, by choosing suitably designed parameters. For example, here is a plot of the pdf when $n= 4$, and $\mu _1\to 4,\mu _2\to 3,\mu _3\to 2,\mu _4\to 7$ and $\sigma _1\to 0.1,\sigma _2\to 2,\sigma _3\to 3,\sigma _4\to 4$:

enter image description here

More usual would just be a skewed right pdf.

Notes 1. As disclosure, I should add that I am one of the authors of the function used above.

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If we start with the CDF, \begin{align*} F_Z(z) &= P(Z \leq z) \\ &= P(\max(X_1,X_2,...,X_N) \leq z) \\ &= P(X_1 \leq z, X_2 \leq z,...,X_N \leq z) \\ &\stackrel{(ind)}{=} \prod_{1}^N P(X_i \leq z) \\ &= \prod_1^N F_{X_i}(z) \end{align*} Taking the derivative yields the pdf. It doesn't seem to me like this will lead to a nice general expression.

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Ross 2003 gives an upper bound for the more general case, where $X_i$ can be dependent:

$$\mathbb{E} \left[ \max_i X_i \right] \le c + \sum_{i=1}^N \left(\sigma^2_i + f_i(c) + (\mu_i - c) Pr\left\{X_i > c \right\} \right)$$ for all values of $c$, where $f_i(c)$ is the pdf of $X_i$.

The tightest bound can be obtained by using a $c$ that satisfies

$$\sum_{i=1}^N Pr\left\{X_i > c \right\} = 1 \: . $$

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