True or False: Joint CDF Has Only One Global Max?

[crbazevedo]

True or false: "Every joint CDF has only one global maximum at F(x1*, ..., xn*) = 1?

I know that the multivariate CDF is monotonically non-decreasing in each of its variables. But does that imply that it has only one global maximum? Is it possible to have two or more separate peaks where the densities sum to one, given that there is no total order in a multidimensional space R^n? I'd guess the answer to this last question is "yes", but I can't figure it out by my own. Thanks.

 

[mathman]

There is only one max, when all arguments become infinite. It is of course possible to achieve this earlier, but it can't go downhill from there - so the function could have a sort of plateau shape.

Example: F(x,y)
Theorem: assume x1 < x2 and y1 < y2, then F(x1,y1) ≤ F(x2,y2).
Proof: F(x1,y1) ≤ F(x1,y2) ≤ F(x2,y2)

 

[crbazevedo]

The existence of plateaus is something I had noticed empirically before. Your example confirms this, what is great, thanks.

Now I'm wondering whether it is possible for a continuous joint CDF to have a discrete set of k separate maxima, say {(x1*,y1*), ..., (xk*,yk*)}, each of which yelding F(x1*,y1*) = ... = F(xk*,yk*) = 1.

Also, would you point me out any textbooks with examples such like yours? I'm particularly interested in partial order statistics in R^n.

Thanks once again.

 

[crbazevedo]

Well, let me roughly define what I mean by "separate":

Without loss of generality, I say that two points x, y in R are "separated" if there exist 0 < epslon < ||x - y|| in R, so that the intersection between A = {x + epslon, x - epslon} and B = {y + epslon, y - epslon} is empty, where ||x - y|| is the Euclidian norm.

I've badly defined this awkard concept in an attempt of excluding the case of continuous platous. Apologizes if this does not make sense (I'm not a mathematician), but I hope it may convey what I mean by "separate peaks".

Thinking a little more about it, based on @mathman's reply, I now think it's impossible to have n+1 separate global maxima in R^n, because the joint CDF is non-decreasing. But in my mental experiment, I still can visualize a 3D shape corresponding to a bivariate CDF in which it is possible to exist two separate peaks.

Now, if I'm not under the effects of any hallucinogen substances, then, the question would be if this in fact can happen in practice.

 

[mathman]

Any discontinuities in distribution functions can only by jump ups with increasing argument.

If F(x1,y1)=F(x2,y2)=1, then for all x > x1, F(x,y1)=1, etc.

 

[bpet]

If by two separate peaks you mean two separate places where the CDF "first" reaches F=1, no it's not possible.

Suppose that F(x1,y2)=1 and F(x2,y1)=1 but F(x1,y1)<1 where x1<x2 and y1<y2. Since F is non-decreasing we have F(x2,y2)=1. But this implies that P(x1<X<=x2,y1<X<=y2) = F(x2,y2)-F(x1,y2)-F(x2,y1)+F(x1,y1) < 0, a contradiction, so F(x1,y1)=1.