I have a data set which consists of > 500 hedge funds, their historical monthly returns, and their benchmark (index) monthly returns. The number of data points (# of monthly returns) differs from a fund to fund (can be as low as 10 and high as 200).
I want to be able to find hedge funds which provide diversification benefit during the bear market (left tail risk). I don't think I should use Pearson correlation coefficients because there are many assumptions that do not hold true (1. the relationship between variables is linear, 2. the relationship between variables at the extreme market is similar to the less extreme, calm market, 3. the joint distribution is normal, etc.).
After doing some googling, I feel like I have to use something called Copula, or some other methods.
What methods can I use to score investments (hedge funds in this case) based on their diversification benefit during the bear market (left tail risk)?