This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# KuiperTest

 KuiperTest[data] tests whether data is normally distributed using the Kuiper test. KuiperTest tests whether data is distributed according to dist using the Kuiper test. KuiperTest returns the value of .
• 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 or multivariate .
• 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 mean of the univariate marginal test statistics is used. -values are computed via Monte Carlo simulation.
• KuiperTest returns a HypothesisTestData object htd that can be used to extract additional test results and properties using the form htd["property"].
• KuiperTest 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 KuiperTest is then used to estimate the -value.
Perform a Kuiper test for normality:
Test the fit of some data to a particular distribution:
Compare the distributions of two datasets:
Extract the test statistic from a Kuiper test:
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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 Scope   (9)
Perform a Kuiper 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 Kuiper test:
The full test table:
A -value table:
The test statistic:
Retrieve the entries from a Kuiper test table for custom reporting:
Report test conclusions using and :
The conclusion may differ at a different significance level:
 Options   (3)
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:
 Applications   (2)
A power curve for the Kuiper test:
Visualize the approximate power curve:
Estimate the power of the Kuiper test when the underlying distribution is a UniformDistribution, the test size is 0.05, and the sample size is 12:
Spatial signs transform bivariate data to points on a unit circle and are often used in tests of location. The null hypothesis can be tested by determining whether the spatial signs are significantly clustered:
The spatial sign function:
Clustering occurs for nonzero mean vectors:
A bivariate test for location using spatial signs and Kuiper's test:
The test does not reject when the mean is close to the null value:
The test rejects the null hypothesis when the mean is not close to the null value:
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, KuiperTest 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:
Independent marginal densities are assumed in tests for multivariate goodness-of-fit:
The test statistic is identical when independence is assumed:
The Kuiper test is not intended for discrete distributions:
The Kuiper 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:
The distribution of the Kuiper test statistic:
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