# Do M-estimators and L-estimators overlap?

In my self-study, I have read the wikipedia entries and some books in regards to M-estimators and L-estimators.

I understand that M-estimators are so called "M" because they "Maximize" the likelihood function, and/or functions of the data itself. I also understand that L-estimators are so called because they use "Linear" combinations of order-statistics of the data.

It seems to me though that there are overlaps between M-estimators and L-estimators, is this not correct? (For example, the mean seems to me that it can be derived as an M-estimator, or as an L-estimator).

Put another way, if we were to draw a Venn Diagram of all M and L estimators, how much would they overlap with each other, if at all?

Edit:

Thanks to the feedback, I understand that such a Venn Diagram might have infinite elements. What I would like however, is very rough percentages of overlap, based on ones' experience, since I know that this experience is far greater than mine! I am just after a rough estimate based on what experts and practitioners in the field have seen and come across.

Thank you.

There are certainly M-estimators that are L-estimators.

An example: the sample median is both an M-estimator and an L-estimator. It's maximum likelihood for the location parameter of a Laplace (double exponential).

The sample midrange is arguably both an M-estimator and an L-estimator, too. I believe it's the maximum likelihood estimator for the center ($\theta$) of a uniform on $(\theta-\phi/2,\theta+\phi/2)$.

the mean seems to me that it can be derived as an M-estimator, or as an L-estimator

Yes, that's another example.

Put another way, if we were to draw a Venn Diagram of all M and L estimators, how much would they overlap with each other, if at all?

I don't know that this is an especially useful way to look at it; my guess is that the number of estimators in the intersection could well be infinite, but may well be a vanishingly-small fraction of the union. But I think it would be very difficult to actually do this calculation, and I believe most of the estimators contained in it would not be interesting for any real-world application.

What might be of more relevance is the proportion of estimators in wide use (amongst those that are M-, or perhaps L-), that are both. But this then gets into issues of personal opinion about definitions and difficulties of estimation (what's 'wide'? how would we measure this?)

I'd think there's probably quite a few that are both, but as a proportion of say all M-estimators that get used/discussed, probably not that large a fraction.

• Thanks Glen_b. Yes, that last sentence is where I was headed, in regards to the proportion of M- and L- estimators that are in 'common' use. – Creatron Dec 24 '13 at 1:59
• I agree that in that form, it's an interesting question to consider, but we'd have to pin down what we mean ... and then figure out some way to work it out/estimate it. It's either going to be pretty much opinion (if possibly somewhat educated opinion) or a study beyond the scope of the short-answer format. – Glen_b -Reinstate Monica Dec 24 '13 at 2:08
• Glen_b, I would settle for an educated opinion based on your experience. Cheers! – Creatron Dec 24 '13 at 16:38
• If you can see some way to make the question rely more on actual expertise, it's probably okay, but to my eye, any attempt to answer it is likely to reflect cognitive bias more than actual fact. I don't see a way to properly back up any answer I might give. – Glen_b -Reinstate Monica Dec 24 '13 at 16:48
• Glen_b, I understand you, no worries. I have edited the question accordingly. – Creatron Dec 24 '13 at 17:28

Here is a Venn Diagram showing the overlap between M- and L-estimators.

• I figured as much - what (rough) percentage would you say that the intersection has of all the M- and L- estimators that are 'common'? Something rough just to get a first order idea. 10%? 20%? – Creatron Dec 24 '13 at 2:26
• As Glen_b noted, the green part of the diagram could have infinite size. We have not yet exhausted the world of estimators - just as we have not exhausted the world of statistics to be estimated. – alex keil Dec 24 '13 at 2:34
• To be only slightly less unhelpful, my particular (applied) field is mainly interested in estimating means and standard deviations. The mean fits under both, whereas the standard deviation does not. Thus, in this group of two, the overlap is 50%. If we draw in more of the field than just the main set of standard analyses, the number of L-estimators does not grow as quickly as M-estimators. This is coming from a very non-theoretically inclined field, mind you. – alex keil Dec 24 '13 at 2:44
• Alex, I realize that. Like I said, I am not looking for an esoteric theoretical quantity. I am just trying to get an admmitedly rough idea based on estimators that are in common use. – Creatron Dec 24 '13 at 2:54
• Ok, that is a good idea based in your field, cheers for that! – Creatron Dec 24 '13 at 2:55