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:

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.

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: