Let $(X_i)_{1\leq i \leq n}$ and $(Y_i)_{1\leq i \leq n}$ be i.i.d. samples of two different distributions. Then, define the "weighted empirical cumulative distributions functions" $F_{n,Y}(t)=\sum_{i=1}^{n}p_i\mathbb{1}_{Y_i \leq t}$ and $F_{n,X}(t)=\sum_{i=1}^{n}p_i\mathbb{1}_{X_i \leq t}$ for some weights $p_i$ that sum to 1, and denote by $F^{-1}_{n,X}$ and $F^{-1}_{n,Y}$ their associated quantile functions. Finally, assume that for each $i$, $$|X_i-Y_i| \leq \varepsilon_n$$ for some sequence $\varepsilon_n$ that converges to 0 at infinity.
I am under the impression that this implies that for any $\alpha$, $$|F^{-1}_{n,X}(\alpha)-F^{-1}_{n,Y}(\alpha)| \leq 2\varepsilon_n.$$ My intuition is as follows :
- if the ordering between the $X_i's$ and the ordering between the $Y_i's$ are the same, the difference between the quantiles will be directly controlled by $\varepsilon_n.$
- if there is a mismatch on the ordering on some subset $I$ of $\left\{1,\dots,n\right\}$, I think that implies that the $X_i's$ for $i \in I$ must be at most $\varepsilon_n$ apart so that the quantile difference would be bounded by $2\varepsilon_n$ (I know this isn't very clear because I'm struggling to identify the proper argument)