Motivating problem: We have $n$ students writing a mock test, and a day after, they write a final test. Let $X_i$ represent the grade (continuous from $0$ to $\infty$) of $i$-th student from the first test, and $Y_i$ from the second test.
Let $S$ be the set of $k<n$ best students from the mock test, i.e. $S:=\{i\leq n: X_i> X_{(n-k)}\}$. Here, $X_{(i)}$ denotes the i-th order statistic, i.e. $X_{(n)} = max(X_1, \dots, X_n)$.
The question is about the distribution of the final grade from the second test between the students in $S$. In particular, I would like to know the (asymptotic) distribution of $$ \Psi = \frac{1}{k}\sum_{i\in S}F(Y_i), $$ where $F$ is the distribution of $Y_1$.
More detailed mathematical notation: Let $(X_1, Y_1), \dots, (X_n, Y_n)$ be iid random vectors. Assume some statistical model, for example let $$Y_i = X_i +\varepsilon_i, $$ where $\varepsilon_i$ are iid independent of $X_i$. Let $F$ be a df of $Y_1$ and let $k<n$.
My question: What is the distribution (or mean) of $$ \Psi = \frac{1}{k}\sum_{i=1}^nF(Y_i)1[X_i> X_{(n-k)}]? $$ Here, $X_{(i)}$ denotes the i-th order statistic, i.e. $X_{(n)} = max(X_1, \dots, X_n)$.
My approach: Consider the case when $\varepsilon_i = 0$ for all $i$. Then, $\Psi = \frac{1}{k}\sum_{i=1}^kF(Y_{(n-i+1)})=$average of $k$ largest order statistics. Since $F(Y_i)\sim U(0,1)$, this is a sum of beta distributions (since it is well-known that order statistics of uniform variables have beta distribution).
However, if $\varepsilon_i \neq 0$, then we are averaging not nessesarily $k$ largest statistics. Do you have any idea what might help with solving this? I heard about concominants of order statistics (section 6.8 in David, H. A.; Nagaraja, H. N. (2003). Order Statistics. Wiley Series in Probability and Statistics), but I am not sure if there is a theorem that can help with this.