Empirical CDF
The empirical distribution function of a sample,
Fn(y), is the proportion
of observations less than or equal to y.
![F_{n}(y) = \frac{1}n
\sum_{i=1}^n{I( y_{i} {\le} y)}](images/disteq135.gif)
where n is the number of observations,
and
is an indicator
function with value 1 if
and with value 0 otherwise.
The Kolmogorov statistic D is a measure
of the discrepancy between the empirical
distribution and the hypothesized distribution.
![D = \rm{Max}_{y} {| F_{n}(y) - F(y)|}](images/disteq138.gif)
where F(y) is the hypothesized
cumulative distribution function.
The statistic is the maximum vertical distance
between the two distribution functions.
The Kolmogorov statistic can be used to construct a confidence
band for the unknown distribution function, to test for a
hypothesized completely known distribution, and to test for
a specific family of distributions with unknown parameters.
If you select a Weight variable,
the weighted empirical distribution function is the proportion
of observation weights for observations less than or equal to y.
![F_{w}(y) =
\frac{1}{\sum_{i}^{}{w_{i}}}
\sum_{i=1}^n{w_{i} I( y_{i} {\le} y)}](images/disteq139.gif)
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.