I see, well, I can’t figure out how to modify the computation to minimize the risk for all values of 𝜃 at once. So calculus of variations just doesn’t work here?

And do you agree with the computation? I don’t understand how to interpret it.

I realised it is totally unclear what I want to say. Sorry for wasting your time. I completely rewrote the question and added a computation that confused me. Hopefully it is at least clear now what I was trying to do.

Mathematically theta is a parameter. But here an estimator cannot depend on theta. To put it differently, a constant estimator is ‘not allowed’. But how could we express it more formally? For example, it seems like Lagrange multipliers method does’t work here (leads to a constant estimator).

Right, given this fact my question sound a bit stupid. But I don’t care about the bound itself. Let’s say we just consider the problem to find UMVUE. It is basically the optimisation problem I am taking about. The only problem is that mathematically it looks a bit different from what I am used to. In particular, a constant estimator looks like a mathematically correct solution, and that is absurd.

1. I see, thanks. 2. I understand but I think it should be possible to formulate a meaningful opt. problem even if there is no (in general) solution to it. I don’t know to forbid a constant estimator.

@RyanBudney I see. Klein bottle is another good example to consider. So the group $\text{Diff}_+(S^1\times S^1)$ has the homotopy type of $S^1\times S^1$? And $\text{Diff}(K)$ has the homotopy type of $S^1$? Torus is a union of two cylinders, and Klein bottle is a union of two Moebius bands so it seems like we can apply the same technique we used for $\mathbb{R}P^2.$

@Allen: so is it some general phenomena that $\text{Diff}(o(M))$ has the same homotopy type as two copies of $\text{Diff}(M)$ where $o(M) \to M$ is the orientation cover? P.S. Thanks for the answer, I am not sure which answer should be the accepted, I think I will just leave it as is.