@Scortchi Heads-up on a new comment at stats.stackexchange.com/questions/294483/… which seems to miss your point entirely.

The very first example in Cox and Hinkley of sufficiency concerns the Poisson distribution on p.19, which concludes that the sum, or equally the mean, is a sufficient statistic for the Poisson. The thread cited concludes that the median cannot be a sufficient statistic, by showing that such an assumption leads to a contradiction. I am reluctant to reply myself for what (paraphrasing Littlewood in another context) are delicate but sufficient grounds (pun intended).

@Jonathan Search stats.stackexchange.com/search?q=Mann+Whitney+tail+answers%3A1 . This hit might address whatever questions you have: see stats.stackexchange.com/questions/65566 .

@whuber Actually, I couldn't find any answer for my latest question in particular.

I was hoping for someone to shed some light on that.