APPLICABILITY OF TWO-SAMPLE GoF TESTS

GoF Test Distribution Family Description
Anderson-Darling Binned
Unbinned
This test can be applied in any case (binned/unbinned data). For its mathematical formulation, the test statistics is sensitive to the tails of the distributions. It can be applied also in case of fat tails.
This test is one of the much powerful among the ones contained in the GoF Toolkit.
Anderson-Darling
(approximated)
Binned
Unbinned
This test can be applied in any case (binned/unbinned data). For its mathematical formulation, the test statistics is sensitive to the tails of the distributions. It can be applied also in case of fat tails. Its limiting distribution is based on an approximation of the upper and lower tail areas of the Anderson-Darling test statistics.
This test is one of the much powerful among the ones contained in the GoF Toolkit.
Chi-squared
(GSL)
Binned 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.
The p-value calculation uses the GSL algorithm.
This test cannot applied if the countings per bin are lower than five. In this case it is possible to handle the original distribution rebinning data until the minimum theoretical frequency is reached. Otherwise, it is possible to apply Yates' correction to chi-squared test statistics (see description). This correction is not yet contained in the GoF Toolkit.
Chi-squared
(approximated)
Binned 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.
The p-value calculation uses the incomplete Gamma function.
This test cannot applied if the countings per bin are lower than five. In this case it is possible to handle the original distribution rebinning data until the minimum theoretical frequency is reached. Otherwise, it is possible to apply Yates' correction to chi-squared test statistics (see description). This correction is not yet contained in the GoF Toolkit.
Chi-squared
(integrating)
Binned 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.
The p-value calculation uses the Gamma function.
This test cannot applied if the countings per bin are lower than five. In this case it is possible to handle the original distribution rebinning data until the minimum theoretical frequency is reached. Otherwise, it is possible to apply Yates' correction to chi-squared test statistics (see description). This correction is not yet contained in the GoF Toolkit.
Fisz-Cramer-von Mises Binned
Unbinned
This test can be applied in any case (binned/unbinned data). This test is satisfactory in case of symmetric or right-skewed distributions.
This test is one of the much powerful among the ones contained in the GoF Toolkit.
Girone Unbinned This test can be applied on unbinned data. This test is a modified version of Cramer-von Mises statistics.
This test can be applied only if the two samples have the same size.
Goodman Unbinned Being Kolmogorov-Smirnov test approximation to a chi-squared statistics, this test is based on Kolmogorov theorem's assumptions as weel and for this reason it cannot be applied in case of binned data. Also in this case, some books show wrong examples of Goodman test applications.
Among the algorithms contained in the GoF Toolkit, Goodman test has a medium power.
Kolmogorov-Smirnov Unbinned This test derives from Kolmogorov statistics and ts applicability rules are based on Kolmogorov theorem. This theorem states that this test can be applied only on unbinned data; many examples available in literature show some Kolmogorov-Smirnov test applications on binned data, but they are wrong. This test has a generalisation to the case of binned data, but these algorithms are not yet contained in the GoF Toolkit.
Among the algorithms contained in the GoF Toolkit, Kolmogorov-Smirnov test has a medium power.
Kuiper Unbinned This test can be applied on unbinned data. This test can be applied on cyclic observations because the test statistics is invariant on the choice of the origin. The test is both sensitive to the tails and to the median of the distributions.
Among the algorithms contained in the GoF Toolkit, Kuiper test has a medium power.
Tiku Binned
Unbinned
This test can be applied in any case (binned/unbinned data). This algorithm converts Cramer-von Mises test statistics into a chi-squared. As for the moment the GoF Toolkit does not contain the algorithm for the exact p-value computation for Fisz-Cramer-von Mises test statistics, it is possible to compute the exact p-value on Tiku approximation.
This test is one of the much powerful among the ones contained in the GoF Toolkit.
Watson Unbinned This test can be applied on unbinned data. This test can be applied on cyclic observations because the test statistics is invariant on the choice of the origin. The test is both sensitive to the tails and to the median of the distributions.
Among the algorithms contained in the GoF Toolkit, Kuiper test has a medium power.
Weighted Cramer von Mises
(Buning weighting function)
Unbinned This test can be applied on unbinned data. This test is a modified version of Cramer-von Mises test. The mathematical formulation of the test statistics emphasises the lower part of the underlying distributions.
This test is one of the much powerful among the ones contained in the GoF Toolkit.
Weighted Kolmogorov-Smironov
(AD weighting function)
Unbinned This test can be applied on unbinned data. This test is a modified version of Kolmogorov-Smirnov test. The mathematical formulation of the test statistics makes it is sensitive to the tails of the distributions. It can be applied also in case of fat tails.
This test is one of the much powerful among the ones contained in the GoF Toolkit.
Weighted Kolmogorov-Smirnov
(Buning weighting function)
Unbinned This test can be applied on unbinned data. This test is a modified version of Kolmogorov-Smirnov test. The mathematical formulation of the test statistics emphasises the lower part of the underlying distributions.
This test is one of the much powerful among the ones contained in the GoF Toolkit.
  


Barbara.Mascialino@ge.infn.it

Last update: 11 April 2006