POWER OF TWO-SAMPLE GoF TESTS

GoF Algorithm Family Power Description
Chi-squared Test

Chi-squared (GSL)
Chi-squared (approximated)
Chi-squared (integrating)

This test has a very general applicability and for this reason it is also known as "the omnibus test". On the other hand, its generality gives to it very little power; in fact, the chi-squared test is the least powerful test among the ones contained in the GoF Toolkit. This might be expected, since the chi-squared statistics loses information when the test is performed on unbinned data because of the need to group the data into bins.
Supremum Statistics Tests

Kolmogorov-Smirnov test
Goodman test
Kuiper test

The test based on the supremum statistics are in general more powerful than the chi-squared test [Cicchitelli, 2001], [Stephens, 1993]. This could be ascribed to the information loss that cannot be avoided while grouping data in chi-squared test.
In particular, comparing the chi-squared test with the Kolmogorov-Smirnov test, Kac, Kiefer and Wolfowitz [Kac, Kiefer and Wolfowitz, 1955] showed that Kolmogorov test statistics requires n^(4/5) observations compared with n observations for chi-squared statistics to attain the same power. Many Monte Carlo studies have demonstrated this fact, in particular in the case of small samples [Stephens, 1993].
Quadratic Statistics Tests

Anderson-Darling
Fisz-Cramer-von Mises
Girone
Tiku
Watson
Weighted Kolmogorov-Smirnov (AD)
Weighted Kolmogorov-Smirnov (Buning)
Weighted Cramer-von Mises (Buning)

The test based on the quadratic statistics are expected to be superior than the test based on the supremum statistics, since they make the comparison between the two distributions all along the range of x, rather than focusing on the maximum distance in one selected point. These algorithms provide the most powerful tests among the ones contained in the GoF Toolkit [Stephens 1974], [Kendall & Stuart, 1979], [Stephens, 1986] .
  


Barbara.Mascialino@ge.infn.it

Last update: 11 April 2006