# Numerical superiority necessary to beat in $L^\infty$ a population one standard deviation ahead

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 $$\max(X_i)<\max(Y_j)$$?

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.

• 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 $n.$
– whuber
Commented Nov 3, 2021 at 18:30
• @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.
– user225256
Commented Nov 3, 2021 at 18:51
• The utility of that observation lies in showing how to standardize the integral to obtain a suitable asymptotic expression.
– whuber
Commented Nov 3, 2021 at 19:33
• I'd be happy to see an expression suitable for figuring out the asymptotic behavior -- e.g. for which $\alpha$ is $m \sim O(n^\alpha)$?
– user225256
Commented Nov 3, 2021 at 19:44