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KuiperTest
✖
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 Automaticthe method to use for computing -values
    SignificanceLevel0.05cutoff 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

open allclose all

Basic Examples  (4)Summary of the most common use cases

Perform a Kuiper test for normality:

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-e2z9oi
In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-b6snhw
Out[2]=2

Test the fit of some data to a particular distribution:

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-erxs41
In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-dofsl4
Out[2]=2

Compare the distributions of two datasets:

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-b8s07g
In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-ckwifh
In[3]:=3
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-ht6wkq
Out[3]=3

Extract the test statistic from a Kuiper test:

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-m3chh3
In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-cm7ucc
Out[2]=2

Scope  (9)Survey of the scope of standard use cases

Testing  (6)

Perform a Kuiper test for normality:

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-lqxbgq

The -value for the normal data is large compared to the -value for the non-normal data:

In[3]:=3
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-mks25
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-kjd2s
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-budcm8
In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-dtqxca
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-ci5vf
Out[3]=3

Compare the distributions of two datasets:

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-f1whrm
In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-2qof
Out[2]=2

The two datasets do not have the same distribution:

In[3]:=3
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-inhd0f
In[4]:=4
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-bbarau
Out[4]=4

Test for multivariate normality:

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-67n6h
In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-eooal3
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-cn5yan
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-gtwavq
In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-n3thk6
In[3]:=3
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-hlhupv
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-ut5c
Out[4]=4

Create a HypothesisTestData object for repeated property extraction:

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-cdnr2n
In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-bolz27
Out[2]=2

The properties available for extraction:

In[3]:=3
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-e40fsc
Out[3]=3

Reporting  (3)

Tabulate the results of the Kuiper test:

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-cqxopz
In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-be6kk1

The full test table:

In[3]:=3
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-ef4et7
Out[3]=3

A -value table:

In[4]:=4
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-bfqugt
Out[4]=4

The test statistic:

In[5]:=5
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-oi9r56
Out[5]=5

Retrieve the entries from a Kuiper test table for custom reporting:

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-s3qsd
In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-ga3bij
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-gg2n22
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-65k71
Out[4]=4

Report test conclusions using "ShortTestConclusion" and "TestConclusion":

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-bkir6y
In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-cd96sm
In[3]:=3
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-ggy9zn
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-el9mb

The conclusion may differ at a different significance level:

In[5]:=5
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-c53cri
In[6]:=6
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-byyexa
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-bc67bi

Options  (3)Common values & functionality for each option

Method  (3)

Use Monte Carlo-based methods for a computation formula:

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-b56tvj
In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-eyrfe
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-evuhgg
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-xg6xc
In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-499mh

The Monte Carlo estimate converges to the true -value with increasing samples:

In[3]:=3
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-eli8sg
In[4]:=4
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-ba5c5u
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-ccet45
In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-ip8pt1

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

In[3]:=3
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-go8plt
In[4]:=4
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-pfg0ok
Out[4]=4

Applications  (2)Sample problems that can be solved with this function

A power curve for the Kuiper test:

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-f9ry9j
In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-bhp5v
In[3]:=3
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-fyqopk

Visualize the approximate power curve:

In[4]:=4
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-eel2vq
Out[4]=4

Estimate the power of the Kuiper test when the underlying distribution is a UniformDistribution[{-4,4}], the test size is 0.05, and the sample size is 12:

In[5]:=5
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-z6f7c
Out[5]=5

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:

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-crkf30

The spatial sign function:

In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-h23jph

Clustering occurs for nonzero mean vectors:

In[3]:=3
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-bxzy6
Out[3]=3

A bivariate test for location using spatial signs and Kuiper's test:

In[4]:=4
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-bfom3s
In[5]:=5
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-dx71bz
Out[5]=5

The test does not reject when the mean is close to the null value:

In[6]:=6
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-jru3mf
Out[6]=6

The test rejects the null hypothesis when the mean is not close to the null value:

In[7]:=7
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-d7ae2n
Out[7]=7

Properties & Relations  (7)Properties of the function, and connections to other functions

By default, univariate data is compared to a NormalDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-in3dv2
In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-loot1k
In[3]:=3
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-ekgpjw
Out[3]=3

The parameters have been estimated from the data:

In[4]:=4
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-f3jfr4
Out[4]=4

Multivariate data is compared to a MultinormalDistribution by default:

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-dhffw
In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-cddu5a
In[3]:=3
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-hpjfy5
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-bstr67

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

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-d47mlh
In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-io3pky
Out[2]=2

Specified parameters are not estimated:

In[3]:=3
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-hxkimc
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-nx9q2m
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-ob999
In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-ccvtx8
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-lczwnz
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-cyr6pp
In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-cdmjix
Out[2]=2

The parameters are estimated but no correction is applied:

In[3]:=3
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-j8jo0o
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-e87ev

The fitted distribution is the same as before and the -value is corrected:

In[5]:=5
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-voh29
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-bx464
Out[6]=6

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

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-et8fgp
In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-plqmfp
Out[2]=2

The test statistic is identical when independence is assumed:

In[3]:=3
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-fpwagd
Out[3]=3

The Kuiper test works with the values only when the input is a TimeSeries:

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-7py5sj
In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-57bf57
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-vvd88
Out[3]=3

Possible Issues  (3)Common pitfalls and unexpected behavior

The Kuiper test is not intended for discrete distributions:

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-ij76p8
In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-lhq85x
Out[2]=2

The test tends to be quite conservative:

In[3]:=3
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-c4yi92
In[4]:=4
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-btzbk7
In[5]:=5
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-ckrxq
Out[5]=5

Use Monte Carlo methods or PearsonChiSquareTest in these cases:

In[6]:=6
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-lr0kz
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-c82uj5
Out[7]=7

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

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-bfm9lx
In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-bhkslk
Out[2]=2

Provide parameter values if they are known:

In[3]:=3
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-ciyvsi
Out[3]=3

Alternatively, use Monte Carlo methods to approximate the -value:

In[4]:=4
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-fwsfhy
Out[4]=4

Ties in the data are ignored:

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-xhttu
In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-haz18n
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-bj6nv9
Out[3]=3

Differences may be more apparent with larger numbers of ties:

In[4]:=4
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-73kvm
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-di9kaw
Out[5]=5

Neat Examples  (1)Surprising or curious use cases

Compute the statistic when the null hypothesis is true:

In[1]:=1
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-2qqg3c
In[2]:=2
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-ywy3ty

The test statistic given a particular alternative:

In[3]:=3
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-c5cy2n

Compare the distributions of the test statistics:

In[4]:=4
✖
https://wolfram.com/xid/0c0sh6jv3d8w2-87eb6q
Out[4]=4
Wolfram Research (2010), KuiperTest, Wolfram Language function, https://reference.wolfram.com/language/ref/KuiperTest.html.
✖
Wolfram Research (2010), KuiperTest, Wolfram Language function, https://reference.wolfram.com/language/ref/KuiperTest.html.

Text

Wolfram Research (2010), KuiperTest, Wolfram Language function, https://reference.wolfram.com/language/ref/KuiperTest.html.

✖
Wolfram Research (2010), KuiperTest, Wolfram Language function, https://reference.wolfram.com/language/ref/KuiperTest.html.

CMS

Wolfram Language. 2010. "KuiperTest." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/KuiperTest.html.

✖
Wolfram Language. 2010. "KuiperTest." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/KuiperTest.html.

APA

Wolfram Language. (2010). KuiperTest. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/KuiperTest.html

✖
Wolfram Language. (2010). KuiperTest. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/KuiperTest.html

BibTeX

@misc{reference.wolfram_2025_kuipertest, author="Wolfram Research", title="{KuiperTest}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/KuiperTest.html}", note=[Accessed: 19-June-2025 ]}

✖
@misc{reference.wolfram_2025_kuipertest, author="Wolfram Research", title="{KuiperTest}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/KuiperTest.html}", note=[Accessed: 19-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_kuipertest, organization={Wolfram Research}, title={KuiperTest}, year={2010}, url={https://reference.wolfram.com/language/ref/KuiperTest.html}, note=[Accessed: 19-June-2025 ]}

✖
@online{reference.wolfram_2025_kuipertest, organization={Wolfram Research}, title={KuiperTest}, year={2010}, url={https://reference.wolfram.com/language/ref/KuiperTest.html}, note=[Accessed: 19-June-2025 ]}