Anderson-Darling (A-D) Statistic

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See also: Fitting distributions to data, Fitting in ModelRisk, Analyzing and using data introduction

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.