9717

Estimating and Diagnostic Checking in Censored Normal Random Samples

A random sample of size is simulated from either a normal or scaled distribution with mean 100 and standard deviation 15. It is then left-censored corresponding to a detection level , where is the censor rate and is the inverse normal cumulative distribution function. The plot shows the data quantiles plotted against the corresponding quantiles in a normal distribution with mean and standard deviation , where and are estimates that are initially set to and . We imagine a robust fitting line that passes through the bulk of the points. Then adjusting shifts the location of this line while shifts its slope. By adjusting and dynamically using the controls, we can find parameter values for which the hypothetical line lies on the 45° line. In other words, the bulk of the data will lie on the 45° line.
If there are outliers in the data, as is frequently the case with data generated from the scaled distribution, this dynamic graphical method provides a more robust estimation method than Gaussian maximum likelihood and so may produce better estimates of the true parameters and . You can investigate the effect of sample size, censoring, and randomness by varying , , and , respectively. The normal distribution is used in the plot shown in the thumbnail.
This dynamic approach provides not only a robust estimation method but also a diagnostic plot to check the assumption of normality when Gaussian maximum likelihood estimates are used. The approach outlined here can be extended for robust estimation with other distributions, including the log-normal and gamma.

SNAPSHOTS

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

DETAILS

Snapshot 1: by using the maximum likelihood estimates (i.e., setting, and ), we can use this plot as a model diagnostic to check the adequacy of the normality assumption; in this case, we conclude the model is adequate
Snapshot 2: the data was generated from the scaled distribution and is not well fit using the Gaussian MLE
Snapshot 3: a better fit and better estimates of the true parameter values and to the data shown in snapshot 2 are obtained by using the dynamic graph
Gaussian maximum likelihood estimation is often used to estimate the mean and standard deviation in normal random samples [1]. As suggested in [2], the EM algorithm provides an effective algorithm for implementing Gaussian maximum likelihood estimation with censored samples, and this method is used in this Demonstration.
References
[1] M. S. Wolynetz, "Algorithm AS 138: Maximum Likelihood Estimation from Confined and Censored Normal Data," Journal of the Royal Statistical Society. Series C (Applied Statistics), 28(2), 1979 pp. 185–195.
[2] C. R. Robert and G. Casella, Monte Carlo Statistical Methods, New York: Springer, 2004.
    • 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.









 
RELATED RESOURCES
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 © 2014 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+