Consider $n$ iid standard normal random variables $g_1,\ldots,g_n$, and I am interested in the lower bound for the following expectation $$\mathbb{E}\max_{K\subset [n],|K| = k}\sum_{i\in K}g_i^2.$$
This is the sum of $k$ largest chi-squares, and I know this can be calculated since we can get the distribution of order statistics. However, is there a simpler way to compute this?
Any ideas would be appreciated.