I came cross an interesting comment saying
If the dimension of a sufficient statistic $T(X)$ is the same as that of the parameter space, then $T(X)$ is minimal sufficient.
Is this is true? I examined some examples e.g. full-rank exponential families, curved exponential families e.g. $N(\theta, \theta^2)$, and some uniform distributions, e.g. $U(-\theta,\theta)$, $U(0,\theta)$, $U(\theta,\theta+a)$ and found their sufficient statistic has the same dimension with the parameter space and indeed they are minimal sufficient.
Is this quoted statement correct? Can you please provide a reference of a proof, or a counter example?