What are the assumptions (and H0) for Wilcoxon signed-rank test?

Question

I am working on the assumption page of the Wilcoxon signed-rank test (the wilcox "paired test") in Wikipedia.

I was able to locate a reference for the assumption that I wrote there, which are:

Let $Z_i=X_i – Y_i$ for $i = 1, \ldots , n$.

1. The differences $Z_i$ are assumed to be independent.
2. Each $Z_i$ comes from the same continuous population, and is symmetric about a common median θ .
3. The values which $X_i$ and $Y_i$ represent are ordered (at least the ordinal level of measurement), so the comparisons "greater than", "less than", and "equal to" are useful.

However, I get this nagging feeling that I am missing something here. Can anyone correct/expand on this?

Thanks.

Answer 1

Assumption 1 is needed. Assumption 3 is not strong enough. You need X and Y to be on scales that make differences orderable, which can mean that X and Y are interval scaled. Regarding the distributional assumption this depends on how you state the hypothesis. If you want to make an inference about the mean difference (and perhaps about the median?) then you assume the distribution of the differences is symmetric. If you want to test the hypothesis that the probability that the sum of a randomly chosen pair of differences exceeds zero is 0.5 then no distributional assumption is needed.