Overview of Goodness of Fit Tests

The table below provides an overview of the Goodness of Fit (GoF) tests available in the Statistical Toolkit; please refer to the User Documentation and the associated publications for further details. 

GoF test Type of distribution Class Notes on applicability
Anderson-Darling Binned
Unbinned
AndersonDarlingBinnedComparisonAlgorithm
AndersonDarlingUnbinnedComparisonAlgorithm
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.
Anderson-Darling approximated Binned
Unbinned
AndersonDarlingBinnedApproximatedComparisonAlgorithm
AndersonDarlingUnbinnedApproximatedComparisonAlgorithm
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.
Chi-squared Binned Chi2ComparisonAlgorithm It cannot applied if the counting per bin is lower than 5.
Chi-squared
(Incomplete Gamma function)
Binned Chi2ApproximatedComparisonAlgorithm It cannot applied if the counting per bin is lower than 5.
Chi-squared
(Gamma function)
Binned Chi2IntegratingComparisonAlgorithm It cannot applied if the counting per bin is lower than 5.
Fisz-Cramer-von Mises Binned
Unbinned
CramerVonMisesBinnedComparisonAlgorithm
CramerVonMisesUnbinned
ComparisonAlgorithm
This test is satisfactory in case of symmetric or right-skewed distributions.
Girone Unbinned GironeComparisonAlgorithm 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 KolmogorovSmirnovApproximatedComparisonAlgorithm It is the approximation of Kolmogorov-Smirnov test to a chi-squared statistics. Since it is based on the assumptions of Kolmogorov theorem, it may be applied to unbinned data.
Kolmogorov-Smirnov Unbinned KolmogorovSmirnovComparisonAlgorithm It derives from Kolmogorov statistics; since it is based on the Kolmogorov theorem, it may be applied to unbinned data.
Kuiper Unbinned KuiperComparisonAlgorithm It  can be applied on cyclic observations, because the test statistics is invariant with respect to the choice of the origin.
It is  sensitive both to the tails and to the median of the distributions.
Tiku Binned
Unbinned
TikuBinnedComparisonAlgorithm
TikuUnbinnedComparisonAlgorithm
This algorithm converts the Cramer-von Mises test statistics into a chi-squared.
Watson Unbinned WatsonComparisonAlgorithm It  can be applied on cyclic observations, because the test statistics is invariant with respect to the choice of the origin.
Weighted Cramer von Mises
(Buning weighting function)
Unbinned WeightedCramerVonMisesBuningUnbinnedComparisonAlgorithm 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.
Weighted Kolmogorov-Smirnov
(AD weighting function)
Unbinned WeightedADKolmogorovSmirnovComparisonAlgorithm 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.
Weighted Kolmogorov-Smirnov
(Buning weighting function)
Unbinned WeightedBuningKolmogorovSmirnovComparisonAlgorithm 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.

More detailed statistical documentation can be retrieved in:

Introduction to Goodness of Fit tests 
GoF algorithms
Applicability of GoF tests
Power of GoF Tests

and in the publications associated to the Statistical Toolkit.


Last modified 10 April 2006 - Maria Grazia Pia