I've also post this question on Stats Stackexchange as advised in the comment.
Suppose $X=(X_1,X_2,\ldots,X_n)$ are non-negative and have a joint probability density $$\frac{1}{\sigma^n}f\bigl(\frac{x}{\sigma}\bigr)=\frac{1}{\sigma^n}f\bigl(\frac{x_1}{\sigma}, \frac{x_2}{\sigma},\ldots, \frac{x_n}{\sigma}\bigr),$$ where $f$ is known, and $\sigma>0$ is the only unknown parameter. In classical decision theory, we may choose an estimator $d$ to minimize the risk $$\mathbb{E}[L(\sigma,d)|\sigma],$$ here $L(\sigma,d)$ is the loss function. For example, Stein’s loss $L(\sigma,d)=d/\sigma-\log(d/\sigma)-1$. Then the minimum risk estimator can be written as $$d=X_{1}/\mathbb{E}[X_{1}|\sigma=1,Z],$$ where $Z=(Z_1,\ldots,Z_n)$ with $Z_i=X_i/X_n$ is the maximal invariant.
With the different choice of loss function, we may get a different minimum risk estimator. So far they are just some well-established results in the theory of point estimation. Usually, in large sample, we prefer MLE since $f$ is known and MLE is the most efficient $\sqrt{n}$-consistent estimator in this case. I’m just curious about the limiting behavior of above kind of estimators. Intuitively, it is consistent. But what is the convergence rate and limiting distribution? I’ve searched for literature but got no general results about this question.
I know the limiting behavior depends on $f$, in a special case that $f$ is normal, then above estimator coincides with MLE. Will above estimator sometimes have a faster/slower convergence rate than MLE if $f$ satisfies certain conditions? Any references or ideas are welcome!