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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?

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    $\begingroup$ 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. $\endgroup$ – StubbornAtom Nov 30 '19 at 20:55
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The cited paper by David Pollard indeed analyses an example due to Kagan and Shepp [The American Statistician 59 (2005) 54–56]. That example gives a statistic which is not sufficient, but still the Fisher information based on the insufficient statistic is equal to the Fisher information based on the complete data.

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.

So the conclusion, this is not an iff condition. But the cited paper continues to state and prove a theorem 7, which gives a necessary regularity condition for iff to hold.

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