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Suppose $m$ independent random variables $X_i$ have the distribution $\mathcal{N}(0, 1)$, and $n$ independent random variables $Y_j$ (also independent of the $X_i$) have the distribution $\mathcal{N}(1, 1)$.

For which $m$ and $n$ is the event $\max(X_i)\ge\max(Y_j)$ more likely than $\max(X_i)<\max(Y_j)$?

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We get an analytical expression for the probability from

\begin{align} P[x \ge \max (X_i)] & = \Phi (x)^m \\ P[x = \max (X_i)] & = m\Phi (x)^{m - 1}\phi (x) \\ P[x = \max (X_i) \ge\max (Y_j)] & = m\Phi (x)^{m - 1}\phi (x)\Phi (x - 1)^n\\ P[\max (X_i) \ge\max (Y_j)] & = \int_{-\infty}^{\infty} m\Phi (x)^{m - 1}\phi (x)\Phi (x - 1)^n dx \end{align}

I don't see how to simplify that integral, or figure out its asymptotic behavior.

In any case, this gives the following table of $n$'s with the minimal $m$ for which $\max (X_i) \ge\max (Y_j)$ is more likely than $\max (X_i) < \max (Y_j)$: \begin{matrix} m & n \\ 4 & 1\\ 11 & 2\\ 29 & 4\\ 78 & 8\\ 205 & 16\\ 535 & 32\\ \end{matrix}

Update: The data for $n \le 51$ is now in the Online Encyclopedia of Integer Sequences as A348913.

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  • $\begingroup$ The asymptotic distributions of the two maxima are given at stats.stackexchange.com/questions/105745. From these you can determine the asymptotic probability for simultaneously large $m$ and $n.$ $\endgroup$
    – whuber
    Commented Nov 3, 2021 at 18:30
  • $\begingroup$ @whuber, that gives a formula of the form Integrate[(1 - CDF[GumbelDistribution[a, b], y]) PDF[GumbelDistribution[c, d], y], {y, -Infinity, +Infinity}], but how would we get the asymptotic behavior of that? Even for $a=c=0$, $b=1$, $d>0$, Mathematica doesn't give a nice expression for the integral. $\endgroup$
    – user225256
    Commented Nov 3, 2021 at 18:51
  • $\begingroup$ The utility of that observation lies in showing how to standardize the integral to obtain a suitable asymptotic expression. $\endgroup$
    – whuber
    Commented Nov 3, 2021 at 19:33
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    $\begingroup$ I'd be happy to see an expression suitable for figuring out the asymptotic behavior -- e.g. for which $\alpha$ is $m \sim O(n^\alpha)$? $\endgroup$
    – user225256
    Commented Nov 3, 2021 at 19:44

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