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I’m trying to learn about copulas. Two definitions I’ve come across are the *H-volume* of a rectangle, which is defined as $V_H=H(x_2,y_2)-H(x_2,y_1)-H(x_1,y_2)+H(x_1,y_1)$.

The function H, whose domain is a subset of $\mathbb{R}^2$ and whose range is a subset of $\mathbb{R}$, is said to be *2-increasing* if $V_H\geq 0$ for all rectangles in the domain of H.

Even though I understand this concept mathematically, can someone give me a graphical/visual interpretation? Is there an easy way to explain, in words, what a 2-increasing function is?

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In the context of probability, I think your $H$ would be the cumulative probability distribution

$$H(x,y) = P(X < x, Y < y)$$

where $X$ and $Y$ are two random variables you’re interested in. This is sometimes called the copula of the variables $X$ and $Y$.

So if you imagine the $x$-$y$ plane, then the probability that $(X,Y)$ lands in the region of points

$$\{(x,y)\mid x<x_2,y<y_2)\}$$

is given by $H(x_2,y_2)$. If you draw x-y axes centered at $(x_2,y_2)$ then you can visualize this as the probability of landing in the bottom left quadrant.

So if you visualize these “bottem left quadrants” centered at each point $(x_i,y_j)$, which make up a rectangle, then see if you can get the probability of landing in the rectangle by using only the probabilities of being in these “lower-left quadrants”. You’ll get the sum you wrote above (you’ll see in this sum points in the rectangle get counted once, while things outside get counted a total of zero times).

Note you need $(x_2,y_2)$ to be in the upper right of the rectangle. The order of your $(x_i,y_j)’s$ matter. The 2-increasing definition won’t make sense if you’re allowed to move the order of the points around.

In this case, where $H$ is a cumulative probability distribution, you automatically get as a consequence that it is 2-increasing. You can see that $H$ has gradients pointing northeast everywhere, so it’s increasing as you go right and up.

I’m realizing you may be more interested in general 2-increasing functions…I don’t know if you can see it the same way in general (any $f(x,y)$ a linear functional, like $f(x,y)=-x-y$ is 2-increasing). If you take the rectangle to be really small, you can see that the partials $(d/(dx\,dy))H \geq 0$, but I don’t know if there are any more general characterizations.

It might help to re-write your equation like this:

$V_H=[H(x_1,y_1)+H(x_2,y_2)]-[H(x_1,y_2)+H(x_2,y_1)] \geq 0$, or even better,

$\frac{1}{2}\big[H(x_1,y_1)+H(x_2,y_2)\big] \geq \frac{1}{2}\big[H(x_1,y_2)+H(x_2,y_1)\big]$. This says that the line from $H(x_1,y_1)$ (the lower left corner) to $H(x_2,y_2)$ (the upper right corner) is at least as high as the line connecting the other two corners, at the point in the plane where their projections cross.

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