# Sum of the $N-K$ largest out of $N$ normal random variables with different variables

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 $$\sum_{i=1}^{N-K}X_{(i)}$$?