Fisher vs. Asymptotic Consistency - Example using a single observation as the population mean estimator

Question

I am learning about Fisher Consistency and came across this section of a Wikipedia article (https://en.wikipedia.org/wiki/Fisher_consistency#Relationship_to_asymptotic_consistency_and_unbiasedness) which gives the following example of "Fisher consistent but not asymptotically consistent":

Take a sequence of Fisher consistent estimators $S_n$, then define $T_n = S_n$ for $n < n_0$, and $T_n = S_{n_0}$ for all $n ≥n_0$. This estimator is Fisher consistent for all n, but not asymptotically consistent. A concrete example of this construction would be estimating the population mean as X1 regardless of the sample size.

I see how using $X_1$ as the population mean estimator is not asymptotically consistent, but why is it Fisher Consistent?

The loose definition of Fisher Consistent from the same Wikipedia article is "if the estimator were calculated using the entire population rather than a sample, the true value of the estimated parameter would be obtained", and I don't see how $X_1$ achieves this.

Answer 1

According to the way I learnt about Fisher consistency (following the early robustness theory work of Huber and Hampel; but the term may be handled differently elsewhere), the term regards a functional that is defined on a probability measure (e.g., the expected value functional rather than the data mean). Such functionals are not estimators, however they are connected to estimators, as they can be evaluated on empirical distributions, and therefore on data sets. This actually seems in line with the Wikipedia entry, however the handling of the term is slightly different, because "my" definition starts from the functional (from which you can always derive an estimator as long as the functional is globally defined), whereas Wikipedia starts from the estimator (but not for every estimator there is such a functional).

The way I learned to use the term, we can only talk about the Fisher consistency of an estimator in connection to the functional that yields the estimator when applied to the empirical distributions. The first paragraph on Wikipedia under "Relationship to asymptotic consistency and unbiasedness" gives two counterexamples in which the functional is not specified. I'm not sure whether such a functional can be defined, I doubt it. (How can a functional that has the empirical distribution as input know that it should pick out the $X_1$-value?) Now according to my (i.e., Huber's) use of the terminology, as long as there is no functional, I'd say that Fisher consistency is not applicable to these estimators (not all estimators can be written as a functional evaluated on the empirical distribution) rather than that they are "not Fisher consistent".

Ultimately I suspect that this is what they mean. It's not that you can show that for the corresponding functional Fisher-consistency does not hold, rather there is no corresponding functional.