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Iosif Pinelis
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From the context, it appears that by "distribution function" the OP meant the pdf (usually, the distribution function is understood as the cdf). Let $F$ and $G$ denote the cdf's of $X$ and $Z_1$, respectively, so that $F=G^n$. (Note that $1-(1-G)^n$ is the cdf of the minimum, not the maximum, of $Z_1,\ldots,Z_n$. Thus, ofer zeitouni actually showed that the pdf of $\min\{Z_1,\ldots,Z_n\}$ is log concave.)

Let us show that the pdf $f=F'$ of $\max\{Z_1,\ldots,Z_n\}$ is log concave. We have $f=F'=nG^{n-1}g$, where $g:=G'$, the standard normal pdf. Next,
\begin{equation} G^2(\ln f)''=-(n-1)gh-G^2,\quad\text{where } h(x):=xG(x)+g(x). \end{equation} Moreover, $h''=g>0$ and $h(-\infty+)=0=h'(-\infty+)$. So, $h>0$ and hence $(\ln f)''<0$, on the entire real line. Thus, the pdf $f$ is indeed log concave.

Iosif Pinelis
  • 127.7k
  • 8
  • 107
  • 229