Let $x^{(i),n}$ be the $i$-th order statistic of $n$ draws from some distribution function $F$ on $[0,1]$.
I have a claim that $$\mathbb{E}[\max \{ x^{(2),n}, m\}] < \mathbb{E}[\max \{ x^{(1),n-1}, m\}]. $$ I was not able to prove this for general distributions yet.
Does someone have a counterexample? A cdf $F$ where the inequality does not hold?
\begin{align} \int_m^1 x n (n-1) F(x)^{n-2} (1-F(x)) f(x) dx + n F(m)^{n-1} (1-F(m))m > \\ \int_m^1 x (n-1) F(x)^{n-2} f(x) dx + F(m)^{n-1} m \end{align}