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The thread "Are inconsistent estimators ever preferable?" and @whuber's answer in it shows that there exists an inconsistent estimator that can outperform a reasonable consistent one for all finite $n$, for a suitable loss function. @whuber's idea for constructing an example of interest is based on finding a loss function that is minimized not at the true parameter value but elsewhere.

I would like to take the problem one step further. In examples similar to @whuber's, there may exist a pseudo-true (for lack of a better term) parameter value that minimizes the expected value of the loss function. (This value depends on the loss function and the actual parameter value, I think.)

Question: If we have a reasonable consistent estimator for the pseudo-true value (corresponding to a given loss function and the actual parameter value), are there examples of an inconsistent estimator which outperforms it for all finite $n$ with respect to the same loss function?

The thread "Are inconsistent estimators ever preferable?" and @whuber's answer in it shows that there exists an inconsistent estimator that can outperform a reasonable consistent one for all finite $n$, for a suitable loss function. @whuber's idea for constructing an example of interest is based on finding a loss function that is minimized not at the true parameter value but elsewhere. (Edit: the last sentence is incorrect.)

I would like to take the problem one step further. In examples similar to @whuber's, there may exist a pseudo-true (for lack of a better term) parameter value that minimizes the expected value of the loss function. (This value depends on the loss function and the actual parameter value, I think.)

Question 1: If we have a reasonable consistent estimator for the pseudo-true value (corresponding to a given loss function and the actual parameter value), are there examples of an inconsistent estimator which outperforms it for all finite $n$ with respect to the same loss function?

Edit: I misread @whuber's example. There, the loss function is minimized not only elsewhere but also at the true parameter value, contrary to my initial understanding. This makes my question lose ground. However, I have a related question.

Question 2: Let us restrict the choice of loss functions so that they achieve minimum at the true parameter value but not anywhere else. (This rules out the type of loss functions used by @whuber.) If we have a reasonable consistent estimator, are there examples of an inconsistent estimator which outperforms it for all finite $n$?

The thread "Are inconsistent estimators ever preferable?" and @whuber's answer in it shows that there exists an inconsistent estimator that can outperform a reasonable consistent one for all finite $n$, for a suitable loss function. @whuber's idea for constructing an example of interest is based on finding a loss function that is minimized not at the true parameter value but elsewhere.

I would like to take the problem one step further. In examples similar to @whuber's, there may exist a pseudo-true (for lack of a better term) parameter value that minimizes the loss function in population. (This value depends on the loss function and the actual parameter value, I think.)

Question: If we have a reasonable consistent estimator for the pseudo-true value (corresponding to a given loss function and the actual parameter value), are there examples of an inconsistent estimator which outperforms it for all finite $n$ with respect to the same loss function?

The thread "Are inconsistent estimators ever preferable?" and @whuber's answer in it shows that there exists an inconsistent estimator that can outperform a reasonable consistent one for all finite $n$, for a suitable loss function. @whuber's idea for constructing an example of interest is based on finding a loss function that is minimized not at the true parameter value but elsewhere.

I would like to take the problem one step further. In examples similar to @whuber's, there may exist a pseudo-true (for lack of a better term) parameter value that minimizes the expected value of the loss function. (This value depends on the loss function and the actual parameter value, I think.)

Question: If we have a reasonable consistent estimator for the pseudo-true value (corresponding to a given loss function and the actual parameter value), are there examples of an inconsistent estimator which outperforms it for all finite $n$ with respect to the same loss function?

The thread "Are inconsistent estimators ever preferable?" and @whuber's answer in it shows that there exists an inconsistent estimator that can outperform a reasonable consistent one for all finite $n$, for a suitable loss function. @whuber's idea for constructing an example of interest is based on finding a loss function that is minimized not at the true parameter value but elsewhere.

I would like to take the problem one step further. In @whuber's example, there exists a pseudo-true (for lack of a better term) parameter value that minimizes the loss function. (This value depends on the loss function and the actual parameter value, I think.)

Question: If we have a reasonable consistent estimator for the pseudo-true value (corresponding to a given loss function and the actual parameter value), are there examples of an inconsistent estimator which outperforms it for all finite $n$ with respect to the same loss function?

The thread "Are inconsistent estimators ever preferable?" and @whuber's answer in it shows that there exists an inconsistent estimator that can outperform a reasonable consistent one for all finite $n$, for a suitable loss function. @whuber's idea for constructing an example of interest is based on finding a loss function that is minimized not at the true parameter value but elsewhere.

I would like to take the problem one step further. In examples similar to @whuber's, there may exist a pseudo-true (for lack of a better term) parameter value that minimizes the loss function in population. (This value depends on the loss function and the actual parameter value, I think.)

Question: If we have a reasonable consistent estimator for the pseudo-true value (corresponding to a given loss function and the actual parameter value), are there examples of an inconsistent estimator which outperforms it for all finite $n$ with respect to the same loss function?