I know that a complete sufficient statistic $T$ is such that 1) $T$ is sufficient for $\theta$, unknown parameter and 2) $T$ is complete.
So, is it always the case? If the answer is not, what could be an example of a complete but not sufficient statistic? Instead is it true in exponential family distributions?
For completeness you need to consider only the distribution of the statistic $T$ (more correctly, the family of distributions indexed by $\theta$), whereas for sufficiency you need to consider the complete likelihood for $\theta$ as a function of the sample $X$; so it's trivial to come up with examples where $T$ isn't sufficient but is still complete:
The constant statistic $T(X)=7$.
Throw some of your data away & calculate your (previously complete & sufficient) test statistic on what's left. Suppose you've four observations $X_1,\ldots,X_4$ from a Gaussian with mean $\theta$ (& known standard deviation): $T(X)=\frac{X_1+X_2}{2}$ isn't sufficient for $\theta$ but is complete.
Twice the sample mean from a random variable uniformly distributed between $0$ & $\theta$, $T(X)=\frac{2\sum_{i=1}^n X_i}{n}$ (the method-of-moments estimator), is complete, but we know the sufficient statistic for $\theta$ is the sample maximum $X_{(n)}$.
See Complete sufficient statistic for a more interesting example—of a sufficient statistic that isn't complete.
