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Let's assume we have two distributions: Y and Z.

How can we compare P(x<= 0) in these two distributions?

For example let's say P_Y(x<= 0)=.5 and P_Z(x<= 0)=.65, is there anyway to test if P_Z(x<= 0) is significantly larger than P_Y(x<= 0)?

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It sounds like you're looking for a quantile test. A quick search led to this: https://www.jstor.org/stable/2673594?seq=1

I'm sure there are other tests in this category.

In your case, the null hypothesis is that the 50th quantile is equal under Y and Z. A two-sided test would show if the 50th quantile of Z is significantly larger or smaller than that of Y. If Z's 50th quantile is lower, then P_Z(x<= 0) would be larger than 0.5.

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This amounts to a two-sample binomial test of the two binary variables $Y\le 0$ and $Z\le 0$.

Of key importance is that the location (0) is prespecified rather than looking at the curves, pick the position where the difference is large and then hurray about significance.

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  • $\begingroup$ exactly, that's the point, the location (0) is pre-specified because we want to test the difference at that point not the point with the largest difference. $\endgroup$
    – Barbaletta
    May 3 '20 at 18:41

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