Comparative overview of Goodness of Fit Tests

The table below provides a comparative overview of the Goodness of Fit (GoF) tests available in the Statistical Toolkit, as a guidance to the practical usage of the Statistical Toolkit. 

GoF test Type of distribution Class Power Notes on applicability
Anderson-Darling Binned
Unbinned
AndersonDarlingBinnedComparisonAlgorithm
AndersonDarlingUnbinnedComparisonAlgorithm
High 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 Low It has a very general applicability; on the other hand, its generality gives it little power.
It cannot applied if the countings per bin are lower than 5.
Fisz-Cramer-von Mises Binned
Unbinned
CramerVonMisesBinnedComparisonAlgorithm
CramerVonMisesUnbinned
ComparisonAlgorithm
High This test is satisfactory in case of symmetric or right-skewed distributions.
Note: the current release implements tables for the evaluation of p-value; the algorithm for the exact p-value computation of the test statistics will be implemented in future release. The exact p-value can be computed in the Tiku approximation.
Goodman Unbinned KolmogorovSmirnovApproxComparisonAlgorithm Medium 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 Medium It derives from Kolmogorov statistics; since it is based on the Kolmogorov theorem, it may be applied to unbinned data.
Kuiper Unbinned KuiperComparisonAlgorithm Medium 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
High This algorithm converts Cramer-von Mises test statistics into a chi-squared.
Note: while the current Statistical Toolkit release does not contain the algorithm for the exact p-value computation for Fisz-Cramer-von Mises test statistics yet, it is possible to compute the exact p-value in the Tiku approximation.

More detailed statistical documentation can be retrieved in:

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


Last modified 26 March 2004 - Maria Grazia Pia