Multivariate Survival and Distribution Functions

#1
Hello,

I would like to confirm if the attached relationship between the multivariate survival and the distribution functions is correct. I read elsewhere that in the multivariate case, the equality in the attachment isn't true. Am I missing something?

Thank you.
 

BGM

TS Contributor
#2
In general it is incorrect. The equality does not hold in 2nd and 3rd step. The reason is very simple - just look at the attached graph illustrating the idea, in bivariate case.

The two sets are not the complement of each other (see the union of the two regions does not cover the whole plane) and therefore in general the corresponding probabilities do not add up to 1.