---
tag: 4503R8BVF8
kind: concept
name: M-estimator
msc: [62F10, 62F35, 62G35]
specializes: [0SSS6ERQ0S]
sources:
  - url: https://en.wikipedia.org/wiki/M-estimator
same_as:
  wikidata: Q475208
  wikipedia: M-estimator
---

## Summary

An M-estimator is an estimator defined as the minimizer of a sum of a loss function over the observations.
Least squares, least absolute deviations and maximum likelihood are all M-estimators.
The class was introduced in robust statistics, where the loss is chosen to limit the influence of outliers.
A single general theory gives consistency and asymptotic normality for the whole class.

## Explanations

### L1 human

Choose a way to measure how badly a candidate value $\theta$ fits one observation, for example the squared distance $(x - \theta)^2$.
Add up these penalties over all observations and take the $\theta$ with the smallest total.
With squared distance you get the sample mean; with absolute distance $\abs{x - \theta}$ you get the sample median.
Other penalties, which grow more slowly for far-away points, give estimates that are less disturbed by outliers.