Test whether expected shortfalls of two distributions are equal I have paired samples of size 1000 from two distributions. I would like to test a null hypothesis that the 2.5% expected shortfalls1 of the two distributions are equal. How can I do that?
(This is a special case of another problem that deals with overlapping observations and some additional complications.)
1 $q$% expected shortfall is (the negative of) the expected value of the observations belonging to the left tail that is cut off at the $q$% quantile level. Synonyms of expected shortfall are conditional value at risk (CVaR), average value at risk (AVaR), expected tail loss (ETL), and superquantile.

If you can assume that the two distributions are roughly normal or lognormal, this would be my approach.
Assume that $\ln(X)$ and $\ln(Y)$ are binormally distributed with means $m,n$, correlation $r$, and standard deviations $s,t$. Then there are formulas $ES(2.5\%, m, s)$ and $ES(2.5\%, n, t)$ for the expected shortfalls.
Meanwhile let $M,N$ be variables for the sample means of $\ln(X)$ and $\ln(Y)$. Let $Means(m,n,r,s,t)$ be the distribution of $M$ and $N$ under the given parameters, which is also binormal. Let $M_0$, $N_0$ be the values of these variables in this dataset.
Similarly let $R,S,T$ be variables for the sample correlation and standard deviations of $\ln(X)$ and $\ln(Y)$. Let $Devs(m,n,r,s,t)$ be the distribution of $R$, $S$ and $T$ under the given parameters, which also has an explicit approximation (recently in papers by Joarder, apparently originally going back to a 1915 paper by Fisher). Let $R_0$, $S_0$ , $T_0$ be the values of these variables in this dataset.
Now let the test statistic be $f(M,N,S,T) = ES(2.5\%, M,S) - ES(2.5\%, N,T)$. We assume that estimates of $M$ and $N$ are roughly independent of estimates of $R$, $S$ and $T$. So a reasonable first null hypothesis is that $f$ is roughly normal with variance the sum of
\begin{align}
Var_{M,N} &=
Var[f(M, N, S_0, T_0)\, |\ M,N \sim Means(M_0, N_0, R_0, S_0, T_0)]\\
Var_{R,S,T}&=Var[f(M_0, N_0, S, T)\, |\, R,S,T \sim Devs(M_0, N_0, R_0, S_0, T_0)]
\end{align}
These variances can be calculated numerically using the formulas above, and the final test is a z-test for $f(M_0, N_0, S_0, T_0)$ under the distribution
$\mathcal{N}(0,Var_{M,N}+Var_{R,S,T})$.
