Setting: N iid normalised random variables ~ N(μ, 1). The normalised log -likelihood is a function-valued statistic. It is: $S(X)= \cfrac{N \mu^2}{2}+ \mu \sum_{i=1}^{N}{Xi} $. It is easy to prove that S is minimally sufficient for $\mu$ but I don't see how to prove it is not complete. I tried to find some functional of S whose expectation would depend uniquely on the known variance...
Thank you for your help!