Given $N$ independent normal random variables, $X_1,X_2,\ldots,X_N$, the ordered sequence is denoted as $X_{(1)}, X_{(2)}, \ldots,X_{(N)}$ where $X_{(1)}\ge X_{(2)}\ge \cdots\ge X_{(N)}$.

Let $X_i\sim N(\mu_i,\sigma^2_i)$, if $\sigma_i=\sigma_j$ for all $i\neq j$, then we can have the exact distribution of $\sum_{i=1}^{N-K}X_{(i)}$ as mentioned in following paper "On the exact distribution of the sum of the largest n − k out of n normal random variables with differing mean values".

I would like to know if there is any result for identical means and different variances, i.e., Suppose $\mu_i=\mu_j$ for all $i\neq j$ and $\sigma_1\ge\sigma_2\ge\cdots\ge\sigma_N$, what is the distribution of $\sum_{i=1}^{N-K}X_{(i)}$?


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