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

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.

• See the related stats.stackexchange.com/questions/173152/… Aug 25, 2021 at 2:00
• @kjetilbhalvorsen Thanks - I saw this but it didn't help me with my question for $X_1$ achieving Fisher consistency. Could you clarify?
– bob
Aug 25, 2021 at 13:47
• Especially because @jbowman in the comments below says that the Wikipedia example is incorrect. Which I agree with, but I only understand the Fisher consistency definition loosely.
– bob
Aug 25, 2021 at 14:14
• Dec 13, 2021 at 21:39

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".