---
tag: 2F5C10ED1A
kind: concept
name: Quantile regression
msc: [62G08, 62J05]
sources:
  - url: https://en.wikipedia.org/wiki/Quantile_regression
same_as:
  wikidata: Q3455889
  wikipedia: Quantile regression
---

## Summary

Quantile regression models a conditional [[quantile|0X60VE98HN]] of a response, such as the conditional median, as a function of covariates.
Where ordinary [[linear regression|41FVC4P5JB]] fits the conditional mean by least squares, quantile regression minimises an asymmetric absolute loss, the check function.
Fitting several quantile levels shows how the whole conditional distribution changes with the covariates, not only its centre.
Koenker and Bassett introduced it in 1978, and the median case is robust to outliers in the response.

## Explanations

### L1 human

The usual regression line tells you the average weight of children at each age.
Quantile regression can tell you instead the weight below which $90\%$ of children of each age fall.
It does this by penalising points above the line nine times as much as points below it, so the line settles where $90\%$ of the points lie below.
With equal penalties on both sides, it gives the median line, which a few extreme points cannot drag far.