I am interested in pointers for the stochastic ordering of continuous dependent variables with common copulas.
I have a vector $X=(X_1,X_2,...,X_N)$ and a vector $Y=(Y_1, Y_2, ..., Y_N)$. X and Y have a common copula C.
Additionally, the joint density is TP 2 in $X_1,..., X_N$ (likewise for Y)
Finally for each j=2,..,N then $X_j \succeq Y_j$ in the likelihood ratio ordering. $X_1$ has the same distribution as $Y_1$.
Letting $S_X$ denote the sum $$S_X=\sum_{j=1}^N X_j$$ and $S_Y$ be defined equivalently. We can say the following (e.g. https://link.springer.com/article/10.1007/s11749-021-00789-5):
- $ X_j|S_X=s $ is non-decreasing in the likelihood ratio order
- $S_X|X_j=x$ is non-decreasing in the likelihood ratio order
It's also the case that $S_X$ dominates $S_Y$ (likelihood ratio ordering implies the usual stochastic order for the j marginals and combining that with the common copula is Theorem 6.B.14 in Shaked & Shanthikumar "Stochastic Orders")
What I am interested in is ranking the following two conditional expectations:
$E[X_1|S_X=s]$ relative to $E[Y_1|S_Y=s]$
as well as
$E[X_1|S_X>s]$ relative to $E[Y_1|S_Y>s]$
My intuition is that if there is positive dependence among the elements and the X vector has larger values in general, then for the same sum we'd expect a smaller value of $X_1$ than we would $Y_1$, but I'm struggling with either proving this directly or finding it in an existing result. I can simulate this for the joint-normal distribution with positive dependence fairly easily & it holds.
Does anybody have any pointers or know if this is a known result somewhere? Is there an additional restriction (e.g. instead of the likelihood ratio ordering is there something that capture the notion of positive dependence in the previous paragraph?) such that it is known?