I'm currently studying statistical inference, and I read about the admissibility of an estimator.
An example is given as a constant estimator $c$. As estimator must have at most the risk of the constant estimator at $\theta=c$, we have that a necessary condition for the admissibility of an estimator is that it achieves 0 loss at $\theta=c$. This gives that the only estimator possible here is $c$, which is itself a constant.
I just find this statement kind of weird. If it's true, then does it mean that any non-constant estimator under mean square loss is inadmissible? As the square loss is the most widely used loss function, the concept of admissibility seems a little bit extra here.
Can someone correct me if I'm wrong? Thanks.