I have $n$ draws $X_1, X_2, \ldots,X_n$ from a random variable $X$ which is continuous and has values between zero and $\overline{x}$. Let $Y(k)$ be a linear transformation of $X$ such that $Y(k)=a(k)X+b(k)$ with $a$ and $b$ being a function of $k$ and $\frac{\partial Y}{\partial k } <0 $. Finally, let $Z_{(i)} (k)$ be the $i$-th lowest order statistic of $Y(k)$.
I am interested how the average of the $k$ lowest expected values of $Y(k)$ compares to the average of the $k+1$ lowest expected values of $Y(k+1)$.
For instance, which conditions must hold such that
(1) $E(\frac{1}{k}\sum^k_{i=1} Z_{(i)}(k) = E(\frac{1}{k+1}\sum^{k+1}_{i=1} Z_{(i)}(k+1))$
or
(2) $E(\frac{1}{k}\sum^k_{i=1} Z_{(i)}(k) < E(\frac{1}{k+1}\sum^{k+1}_{i=1} Z_{(i)}(k+1))$ ?
I believe that for $a=b=0$ the inequality holds and for $X=\overline(x)$ the equality holds. How do I characterize that there might be an intersection earlier?