# Is it true that Fisher information for a statistic and the sample are equal if and only if the statistic is sufficient?

According to https://en.wikipedia.org/wiki/Fisher_information#Sufficient_statistic we have if and only if, but according to https://projecteuclid.org/download/pdfview_1/euclid.imsc/1362751193 we don't. I have not yet read measure theory so I don't really understand the second link.

Could someone give me some clarity on the if part of the theorem? Is it dependent on regularity conditions?

• I also knew it as an iff condition. See e.g. problem 6.4.15 here where the author mentions a consultation with C.R. Rao regarding the result. – StubbornAtom Nov 30 '19 at 20:55

I will not state that example here, but note that it is a mixture distribution where the support of the distribution varies with the unknown parameter $$\theta$$. That is known to cause many hickups.