My goal is to test if ES (CVaR) empirically is a better risk measure than VaR for a set of given variables (assumed underlying distribution, confidence level, sample size) for different asset classes.
As VaR and ES are like apples and oranges, they can not be compared directly. Most backtesting frameworks that I have found test explicitly VaR or ES (I am aware of the difficulties of backtesting ES) and these backtests are not comparable and it is seems hard to infer if e.g ES is superior to VaR from given samples. I assume there must be involved some function that include a common success factor.
Based on this, I am wondering if there are any known approaches for statistically testing (backtests) and comparing VaR and ES?