Let $X_1, ..., X_n$ be $n$ i.i.d. Let let $f$ be their log-concave PDF and $F$ be their CDF.
The nth order statistic $\max_{i=1...n} X_i$ has for CDF $F_n\left(x\right) = \mathbb{P}\left( \max_{i=1...N} X_i < x \right) = F\left( x \right)^n$ and its PDF is $f_n\left(x\right) = n f\left(x\right) F\left(x\right)^{n-1}$
I am interested in the growth of $\left\lbrace 1 - F\left( x_n \right)^n \right\rbrace_n$ when the number of variables $n$ increases. Here, $x_n = \frac{1 - F\left( x_n \right)^n}{f_n\left( x_n \right)}$. Note that $x_n$ is the fixed point of the Mills ratio of $\max_{i=1...n} X_i$.
I have run some simulations and it seems that $1 - F\left( x_{n+1} \right)^{n+1} > 1 - F\left( x_n \right)^n$ but I have failed to prove it.
Is $\left\lbrace 1 - F\left( x_n \right)^n \right\rbrace_n$ really increasing?
$f$ being log-concave, there are properties I have tried to exploit without success:
- $f_n$ and $F_n$ are log concave
- $\frac{1 - F\left( x \right)^n}{f_n\left( x \right)}$ is decreasing in $x$
- $\frac{f_n\left( x \right)}{F\left( x \right)^n}$ is decreaing in $x$
- $x_n$ is the unique fixed point of $\frac{1 - F\left( x \right)^n}{f_n\left( x \right)}$ because $\frac{1 - F\left( x \right)^n}{f_n\left( x \right)}$ is strictly decreasing
- $\left\lbrace x_n \right\rbrace_n$ is increasing
Finally, if the $X_1, ..., X_n$ are uniformly distributed, $1 - F\left( x_n \right)^n$ is increasing and has an explicit form.