The Sampling Distribution of a Sampling Quantile

Conventional regression lets you take the values of independent variables and predict the mean value of a dependent variable. It thus provides a single equation. The relatively new method of "quantile regression" lets you take the values of independent variables and predict the quantile function of the independent variable. It thus provides a family of equations. It could be said that to determine the 0.35 quantile of the independent variable one uses the function , whereas to predict the 0.65 quantile of the independent variable one uses the function .
Just as conventional regression rests on the assumption that the distribution of errors around a prediction is normal, so too there is a normality assumption with quantile regression. More specifically, it is assumed that the error of the quantile functions is normally distributed.
This Demonstration lets you examine this normality assumption that lies behind quantile regression. You select an underlying distribution. You have the choice of picking a normal distribution, a gamma distribution, or a beta distribution. You also parameterize the underlying distribution and select the size of an underlying random sample drawn from this underlying distribution. You then choose derivative random samples from this underlying distribution. You determine both the size of each of these derivative random samples and the number of derivative random samples taken. You also choose the quantile value at which you want each of the derivative random samples evaluated. This produces a quantile sample.
This Demonstration takes the quantile sample, produces a histogram of it, determines the best-fit normal distribution for that quantile sample, the theoretically predicted versus the actual variance of the best-fit normal distribution, and the results of a test for the normality of the quantile sample. Mousing over the last row of the output produces additional information on the fit of the quantile distribution.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


[1] L. Hao and D. Q. Naiman, Quantile Regression, Thousand Oaks, CA: Sage Publications, 2007.
[2] R. Koenker and G. Bassett, Jr., "Regression Quantiles," Econometrica, 46(1), 1978 pp. 33–50.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2017 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+