I am looking for distances between two random variables $X$ and $Y$, or practical estimates for measuring the distance between the i.i.d. observations $(X^1, \ldots, X^T)$ and $(Y^1,\ldots,Y^T)$.
I am aware of the divergences or statistical distances, but they focus on quantifying the dissimilarity in distribution which is sufficient when $X$ and $Y$ are independent, but fail to measure how "correlated" they are otherwise.
Any information is welcomed!
Solved – Distance between random variables
correlationdistancedistributionsrandom variable
Best Answer
Here's a measure that seems to accord with your requirements for the case of monotonic relationships between $X$ and $Y$:
Let $X,Y$ be your sample vectors. Let $S(X,Y)$ be the Spearman Rank Correlation between these two vectors and let $KS(X,Y)$ be the Kolmorogov-Smirnov Statistic between the EDCF of $X$ and ECDF of $Y$.
We can construct the quantity $D(X,Y)=||S(X,Y)|-1|+KS(X,Y)$. Lets analyze the cases:
So, this is more of a "coefficient" or index than a distance, but maybe itll work for you.