Anderson-Darling (A-D) Statistic | Vose Software

Anderson-Darling (A-D) Statistic

The A-D statistic A2n is defined as:

where

n = total number of data points

F(x) = distribution function of the fitted distribution

f(x) = density function of the fitted distribution

Fn(x) = i/n

i = the cumulative rank of the data point

The Anderson-Darling statistic is a sophisticated version of the Kolmogorov-Smirnoff statistic. It is more powerful for the following reasons:

1. Y(x) compensates for the increased variance of the vertical distances between distributions' sK-S2

2. f(x) weights the observed distances by the probability that a value will be generated at that x-values

3. The vertical distances are integrated over all values of x to make maximum use of the observed data (the K-S statistic only looks at the maximum vertical distance).

The A-D statistic is therefore a generally more useful measure of fit than the K-S statistic, especially where it is important to place equal emphasis on fitting a distribution at the tails as well as the main body. On the other hand, it requires a lot more number-crunching and is not therefore very practical to calculate manually.

An alternative equation for the Anderson-Darling statistic is:

The principle behind the Anderson-Darling statistic is that one is fitting a distribution with known parameter values to the data. When the parameters are being estimated from the data a correction needs to be applied, especially for small samples.