APPLICABILITY OF GOODNESS OF FIT TESTS

GoF Algorithm Distribution Family Description

Chi-squared Test 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.
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.

Kolmogorov-Smirnov Test 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.

Goodman Test 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.

Kuiper Test 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.

Anderson-Darling Test 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 quite complex and for the moment the GoF Toolkit contains only the tables for the p-value evaluation. This test is one of the much powerful among the ones contained in the GoF Toolkit.

Fisz-Cramer-von Mises Test 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.Its limiting distribution is quite complex and for the moment the GoF Toolkit contains only the tables for the p-value evaluation. This test is one of the much powerful among the ones contained in the GoF Toolkit.

Tiku Test 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.

Barbara.Mascialino@ge.infn.it

Last update: 03/26/2004

GoF Algorithm	Distribution Family	Description
Chi-squared Test	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. 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.
Kolmogorov-Smirnov Test	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.
Goodman Test	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.
Kuiper Test	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.
Anderson-Darling Test	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 quite complex and for the moment the GoF Toolkit contains only the tables for the p-value evaluation. This test is one of the much powerful among the ones contained in the GoF Toolkit.
Fisz-Cramer-von Mises Test	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.Its limiting distribution is quite complex and for the moment the GoF Toolkit contains only the tables for the p-value evaluation. This test is one of the much powerful among the ones contained in the GoF Toolkit.
Tiku Test	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.