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
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
are estimates that are initially set to
. 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
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
. You can investigate the effect of sample size, censoring, and randomness by varying
, 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.