# “Consistent estimator” or “consistent estimate”?

Question: Are both expressions "consistent estimator" and "consistent estimate" meaningful?

The quote below is intended to be illustrative; however, I am interested in the question above in a general setting.

Reading a paper on econometrics (Krueger, 2001, pp. 8), I noticed an expression "consistent estimate[s]":

Generalized method of moments provides a way to weight the sample moments efﬁciently to derive consistent estimates of the desired population parameters.

which I found a little weird; I was more used to "consistent estimator". I thought it was a typo. I googled the two expressions to see if one clearly dominates. Interestingly, Google gives quite many hits for both: 190 000 for "consistent estimator" and 126 000 for "consistent estimate".

Here is a thread "What is the relation between estimator and estimate?" with a beautiful answer from @whuber. I guess it could be sufficient, but I would appreciate an answer specifically on whether the concept "consistent" applies to both "estimator" and "estimate" or just one of them.

References

• Krueger, Alan B. "Symposium on econometric tools." The Journal of Economic Perspectives 15.4 (2001): 3-10.
• I'd include examples in the post rather than make this thread depend on reading outside sources. It's easy to guess that some of those uses would offend as incorrect; otherwise what I expect is of the flavour that consistent estimates are those produced by a consistent estimator. I am happy to feel such usage to be on one level absurd and lacking rigour or even correctness and on another level an excusable extension of standard terminology. – Nick Cox Feb 11 '16 at 10:02
• @NickCox, done. I should probably have omitted the reference in the first place. I am looking for a general explanation which I suspect to be quite simple (something like what you indicated). – Richard Hardy Feb 11 '16 at 10:49
• Most of us can combine a strong desire to write carefully and correctly with an expectation that some of what we write would astonish others, especially those who understand more and understand more deeply and have even stronger drive for correctness. I've not followed up on the context but the quotation seems a little ambiguous. In general "derive" could mean anything from "do the formal manipulations to get a recipe" to "do the calculations to get some numbers". – Nick Cox Feb 11 '16 at 10:55
• @Tim, I feel that my wish for a concrete answer on the specific question is justified. Even if the answer can be inferred (by a qualified individual) from the other post (which I linked in my post right from the start), why not just spell it out in an answer and make it easier for anyone who ponders upon this question in the future? (And gain reputation points, too.) – Richard Hardy Feb 11 '16 at 11:38

The difference between estimator and estimate was nicely described by @whuber in this thread

an estimator is a definite mathematical procedure that comes up with a number (the estimate) for any possible set of data that a particular problem could produce

Now, quoting Wikipedia

consistent estimator or asymptotically consistent estimator is an estimator—a rule for computing estimates of a parameter $\theta_0$—having the property that as the number of data points used increases indefinitely, the resulting sequence of estimates converges in probability to $\theta_0$. (https://en.wikipedia.org/wiki/Consistent_estimator)

Consistency is a feature of estimator, the estimator that is consistent also returns consistent estimates. Estimate itself cannot be consistent, but the estimates produced by consistent estimator are consistent.

Regarding the quote in your question, it is valid. Saying that the estimates are consistent is another way of saying that they come from consistent estimator.

Consistent estimate does not make sens since estimate is just a single number, so it cannot be consistent. Consistency is a feature of some process, or of parts of something, e.g.

the ​quality of always ​behaving or ​performing in a ​similar way, or of always ​happening in a ​similar way (https://dictionary.cambridge.org/dictionary/english/consistency)

or

1. agreement or accordance with facts, form, or characteristics previously shown or stated
2. agreement or harmony between parts of something complex; compatibility
3. (General Physics) degree of viscosity or firmness
4. the state or quality of holding or sticking together and retaining shape
5. conformity with previous attitudes, behaviour, practice, etc

That is why single estimate cannot be consistent, but multiple estimates can be consistent. If they are, they make their estimator consistent as it's behavior (that produced them) is consistent.

• Another way to say it is an estimator is a random variable, while an estimate is a realization of an estimator. – bdeonovic Feb 12 '16 at 12:11
• @bdeonovic, if you read the linked thread carefully, there is a statement that estimator is not a random variable: The estimator itself is not a random variable: it's just a mathematical function. This is quite something! I still get what you mean, though. @Tim, I am happy to upvote your answer (I will wait for alternative answers before accepting). The last three lines of your post are quite important, IMHO. But what do you think about the quote in my question? Is it appropriate? – Richard Hardy Feb 12 '16 at 12:17
• That is not in line with what I learned about estimators. Wikipedia agrees with me for what thats worth. Also we often talk about "consistency" of estimators and other properties like that which are properties of random variables. I think it is much more common to consider estimators random variables. – bdeonovic Feb 12 '16 at 13:01
• @bdeonovic please refer to the thread that is linked, it describes what is and what is not a RV in detail, you can also always ask whuber for comments on this ;) – Tim Feb 12 '16 at 13:22
• I understand what he is saying, I'm just not agreeing, but its just terminology as far as I am concerned and doesn't really matter. – bdeonovic Feb 12 '16 at 13:29