Let $X_1,X_2... X_n$ be iid $\sim F$ where $F$ is any symmetric continuous distribution and let $\mid E(X)\mid<\infty$.
Is $\bar{x}$ complete sufficient for $E(x)=\int{xf(x)dx}$?
Assume that all parameters of $F$ except $E(x)=\mu$ are known and that the support of $X$ does not depend on $\mu$