BUILT-IN MATHEMATICA SYMBOL

KolmogorovSmirnovTest

KolmogorovSmirnovTest[data]
tests whether data is normally distributed using the Kolmogorov-Smirnov test.

tests whether data is distributed according to dist using the Kolmogorov-Smirnov test.

KolmogorovSmirnovTest

returns the value of

.

MORE INFORMATION

KolmogorovSmirnovTest performs the Kolmogorov-Smirnov goodness-of-fit test with null hypothesis that data was drawn from a population with distribution dist and alternative hypothesis that it was not.

By default a probability value or -value is returned.

A small -value suggests that it is unlikely that the data came from dist.

The dist can be any symbolic distribution with numeric and symbolic parameters or a dataset.

The data can be univariate or multivariate .

The Kolmogorov-Smirnov test assumes that the data came from a continuous distribution.

The Kolmogorov-Smirnov test effectively uses a test statistic based on where is the empirical CDF of data and is the CDF of dist.

For multivariate tests, the mean of the univariate marginal test statistics is used. -values are computed via Monte Carlo simulation.

KolmogorovSmirnovTest returns a HypothesisTestData object htd that can be used to extract additional test results and properties using the form htd["property"].

KolmogorovSmirnovTest can be used to directly give the value of .

Properties related to the reporting of test results include:

	"PValue"	-value
	"PValueTable"	formatted version of
	"ShortTestConclusion"	a short description of the conclusion of a test
	"TestConclusion"	a description of the conclusion of a test
	"TestData"	test statistic and -value
	"TestDataTable"	formatted version of
	"TestStatistic"	test statistic
	"TestStatisticTable"	formatted

The following properties are independent of which test is being performed.

Properties related to the data distribution include:

	"FittedDistribution"	fitted distribution of data
	"FittedDistributionParameters"	distribution parameters of data

The following options can be given:

	Method	Automatic	the method to use for computing -values
	SignificanceLevel	0.05	cutoff for diagnostics and reporting

For a test for goodness-of-fit, a cutoff is chosen such that is rejected only if . The value of used for the and properties is controlled by the SignificanceLevel option. By default is set to .

With the setting Method, datasets of the same length as the input are generated under using the fitted distribution. The EmpiricalDistribution from KolmogorovSmirnovTest is then used to estimate the -value.

EXAMPLES

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Basic Examples (3)

Perform a Kolmogorov-Smirnov test for normality:

Test the fit of some data to a particular distribution:

Compare the distributions of two datasets:

Perform a Kolmogorov-Smirnov test for normality:

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Test the fit of some data to a particular distribution:

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Compare the distributions of two datasets:

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Scope (9)

Perform a Kolmogorov-Smirnov test for normality:

The

-value for the normal data is large compared to the

-value for the non-normal data:

Test the goodness-of-fit to a particular distribution:

Compare the distributions of two datasets:

The two datasets do not have the same distribution:

Test for multivariate normality:

Test for goodness-of-fit to any multivariate distribution:

Create a HypothesisTestData object for repeated property extraction:

The properties available for extraction:

Tabulate the results of the Kolmogorov-Smirnov test:

The full test table:

A

-value table:

The test statistic:

Retrieve the entries from a Kolmogorov-Smirnov test table for custom reporting:

Report test conclusions using

and

:

The conclusion may differ at a different significance level:

Options (4)

Use Monte Carlo-based methods for a computation formula:

Set the number of samples to use for Monte Carlo-based methods:

The Monte Carlo estimate converges to the true

-value with increasing samples:

Set the random seed used in Monte Carlo-based methods:

The seed affects the state of the generator and has some effect on the resulting

-value:

Set the significance level used for

and

:

By default

is used:

Applications (2)

A power curve for the Kolmogorov-Smirnov test:

Visualize the approximate power curve:

Estimate the power of the Kolmogorov-Smirnov test when the underlying distribution is a UniformDistribution

, the test size is 0.05, and the sample size is 12:

A sample of 31 sheets of airplane glass were subjected to a constant stress until breakage. Investigate whether the data is drawn from a NormalDistribution or a GammaDistribution:

Compare the quantile-quantile plots for the candidate distributions:

The data appears to fit a GammaDistribution slightly better than a NormalDistribution:

Properties & Relations (8)

By default, univariate data is compared to a NormalDistribution:

The parameters have been estimated from the data:

Multivariate data is compared to a MultinormalDistribution by default:

The parameters of the test distribution are estimated from the data if not specified:

Specified parameters are not estimated:

Maximum-likelihood estimates are used for unspecified parameters of the test distribution:

If the parameters are unknown, KolmogorovSmirnovTest applies a correction when possible:

The parameters are estimated but no correction is applied:

The fitted distribution is the same as before and the

-value is corrected:

When parameters are estimated Lilliefors' correction is used:

Estimate the parameters prior to testing to perform the classical Kolmogorov-Smirnov test:

Conceptually, the Kolmogorov-Smirnov test computes the maximum absolute difference between the empirical and theoretical CDFs:

Plot the CDFs, showing the maximum absolute difference:

Independent marginal densities are assumed in tests for multivariate goodness-of-fit:

The test statistic is identical when independence is assumed:

Possible Issues (3)

The Kolmogorov-Smirnov test is not intended for discrete distributions:

Use PearsonChiSquareTest instead:

The Kolmogorov-Smirnov test is not valid for some distributions when parameters have been estimated from the data:

Provide parameter values if they are known:

Alternatively, use Monte Carlo methods to approximate the

-value:

Ties in the data are ignored:

Differences may be more apparent with larger numbers of ties:

Neat Examples (1)

The distribution of the Kolmogorov-Smirnov test statistic: