# KuiperTest

KuiperTest[data]

tests whether data is normally distributed using the Kuiper test.

KuiperTest[data,dist]

tests whether data is distributed according to dist using the Kuiper test.

KuiperTest[data,dist,"property"]

returns the value of "property".

# Details and Options

• KuiperTest performs the Kuiper 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 {x1,x2,} or multivariate {{x1,y1,},{x2,y2,},}.
• The Kuiper test assumes that the data came from a continuous distribution.
• The Kuiper 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 sum of the univariate marginal -values is used and is assumed to follow a UniformSumDistribution under .
• KuiperTest[data,dist,"HypothesisTestData"] returns a HypothesisTestData object htd that can be used to extract additional test results and properties using the form htd["property"].
• KuiperTest[data,dist,"property"] can be used to directly give the value of "property".
• Properties related to the reporting of test results include:
•  "PValue" -value "PValueTable" formatted version of "PValue" "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 "TestData" "TestStatistic" test statistic "TestStatisticTable" formatted "TestStatistic"
• 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 "TestConclusion" and "ShortTestConclusion" properties is controlled by the SignificanceLevel option. By default is set to 0.05.
• With the setting Method->"MonteCarlo", datasets of the same length as the input are generated under using the fitted distribution. The EmpiricalDistribution from KuiperTest[si,dist,"TestStatistic"] is then used to estimate the -value.

# Examples

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## Basic Examples(4)

Perform a Kuiper 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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Extract the test statistic from a Kuiper test:

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