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MardiaKurtosisTest
✖
MardiaKurtosisTest

✖

tests whether data follows a MultinormalDistribution using the Mardia kurtosis test.

✖
MardiaKurtosisTest[data,"property"]

returns the value of "property".

Details and Options

  • MardiaKurtosisTest performs the Mardia kurtosis goodness-of-fit test with null hypothesis that data was drawn from a MultinormalDistribution 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 is normally distributed.
  • The data can be univariate {x1,x2,…} or multivariate {{x1,y1,…},{x2,y2,…},…}.
  • The Mardia kurtosis test effectively compares a multivariate measure of kurtosis for data to a MultinormalDistribution.
  • MardiaKurtosisTest[data,dist,"HypothesisTestData"] returns a HypothesisTestData object htd that can be used to extract additional test results and properties using the form htd["property"].
  • MardiaKurtosisTest[data,dist,"property"] can be used to directly give the value of "property".
  • PearsonChiSquareTest[data,dist,"property"] can be used to directly give the value of "property".
  • Properties related to the reporting of test results include:
  • "DegreesOfFreedom"the degrees of freedom used in a test
    "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
    SignificanceLevel 0.05cutoff 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.
  • The following methods can be used to compute -values:
  • Automaticcorrect for small samples up to dimension 5
    "Asymptotic"use the asymptotic distribution of the test statistic
    "MonteCarlo"use Monte Carlo simulation
  • With the setting Method-> "MonteCarlo", datasets of the same length as the input are generated under using the fitted distribution. The EmpiricalDistribution from "MonteCarlo" MardiaKurtosisTest[si,"TestStatistic"] is then used to estimate the -value.

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

Perform a test for multivariate normality:

In[1]:=1
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-ijo5
In[2]:=2
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-hnfx0t
Out[2]=2

Extract the test statistic from the Mardia kurtosis test:

In[1]:=1
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-m3chh3
In[2]:=2
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-cm7ucc
Out[2]=2

Obtain a formatted test table:

In[1]:=1
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-p7hfx
In[2]:=2
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-cr9vsu
Out[2]=2

Scope  (5)Survey of the scope of standard use cases

Testing  (2)

Perform a Mardia kurtosis test for multivariate normality:

In[1]:=1
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-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/0b6g2u59e1po1v6m4hx3k2-mks25
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-kjd2s
Out[4]=4

Create a HypothesisTestData object for repeated property extraction:

In[1]:=1
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-cdnr2n
In[2]:=2
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-bolz27
Out[2]=2

The properties available for extraction:

In[3]:=3
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-e40fsc
Out[3]=3

Reporting  (3)

Tabulate the results of the Mardia kurtosis test:

In[1]:=1
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-cqxopz
In[2]:=2
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-be6kk1

The full test table:

In[3]:=3
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-ef4et7
Out[3]=3

A -value table:

In[4]:=4
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-bfqugt
Out[4]=4

The test statistic:

In[5]:=5
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-oi9r56
Out[5]=5

Retrieve the entries from a Mardia kurtosis test table for custom reporting:

In[1]:=1
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-s3qsd
In[2]:=2
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-ga3bij
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-gg2n22
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-65k71
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-bkir6y
In[2]:=2
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-cd96sm
In[3]:=3
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-ggy9zn
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-el9mb

The conclusion may differ at a different significance level:

In[5]:=5
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-c53cri
In[6]:=6
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-byyexa
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-bc67bi

Options  (4)Common values & functionality for each option

Method  (3)

Use Monte Carlo-based methods or a computation formula:

In[1]:=1
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-b56tvj
In[4]:=4
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-eyrfe
Out[4]=4
In[3]:=3
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-evuhgg
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-xg6xc
In[2]:=2
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-499mh

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

In[3]:=3
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-eli8sg
In[4]:=4
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-ba5c5u
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-ccet45
In[2]:=2
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-ip8pt1

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

In[3]:=3
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-pfg0ok
Out[3]=3

SignificanceLevel  (1)

Set the significance level used for "TestConclusion" and "ShortTestConclusion":

In[4]:=4
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-kj9pa
In[5]:=5
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-qafg
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-bu4h57
Out[6]=6

By default, 0.05 is used:

In[7]:=7
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-b50sx2

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

A power curve for the Mardia kurtosis test:

In[1]:=1
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-f9ry9j
In[2]:=2
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-bhp5v
In[3]:=3
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-fyqopk

Visualize the approximate power curve:

In[4]:=4
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-eel2vq
Out[4]=4

Estimate the power of the Mardia kurtosis test when the underlying distribution is a MultivariateTDistribution, the test size is 0.05, and the sample size is 27:

In[5]:=5
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-z6f7c
Out[5]=5

Measures of petal and sepal dimensions for three varieties of iris were recorded. A multivariate test of means can be used as a quick check that the measures might be useful in discriminating between two similar species but is only valid if the data follows a multivariate normal distribution:

In[1]:=1
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-eenalg
In[2]:=2
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-s0acha
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-ij4e88

The multivariate kurtosis of the two species is similar to a multivariate normal distribution:

In[4]:=4
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-gm520g
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-biwspi
Out[5]=5

The multivariate skewness should also be checked to confirm normality:

In[6]:=6
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-jbz11
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-c2pnp3
Out[7]=7

The data appears normal, so TTest is valid:

In[8]:=8
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-b448mx
Out[8]=8

Properties & Relations  (5)Properties of the function, and connections to other functions

The multivariate test statistic:

In[1]:=1
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-h7l4x6
In[2]:=2
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-dsep4w
In[3]:=3
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-ldjsds
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-etir6z
Out[4]=4

The univariate test statistic:

In[1]:=1
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-ibppux
In[2]:=2
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-d959th
In[3]:=3
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-broj7t
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-ieqkgx
Out[4]=4

The multivariate test statistic has an asymptotic NormalDistribution[0,1]:

In[1]:=1
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-duofhl
In[2]:=2
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-ezrqap
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-clhj8e
Out[3]=3

The asymptotic -value can be very inaccurate for small samples:

In[4]:=4
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-k5txjs
In[5]:=5
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-d6svmq
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-iprgm
Out[6]=6

For comparison, the Monte Carlo -value is much closer to the small-sample value:

In[7]:=7
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-frv7oy
In[8]:=8
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-ff2j5l
In[9]:=9
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-cmetx4
Out[9]=9

Mardia's kurtosis test can only detect departures from normality in kurtosis:

In[1]:=1
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-bx8on9

The data is clearly not normally distributed:

In[2]:=2
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-ds92d7
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-m074ru
Out[3]=3

Decisions should be based on MardiaSkewnessTest and MardiaKurtosisTest:

In[4]:=4
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-e08lru
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-7py5sj
In[2]:=2
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-of2jx
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-57bf57
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-vvd88
Out[4]=4

Possible Issues  (1)Common pitfalls and unexpected behavior

If the covariance matrix of the data is not positive definite, the test will fail:

In[1]:=1
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-e2fnxq
In[2]:=2
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-dmqkla
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-k118o6
Out[3]=3

The number of data points must be greater than the dimension of the data:

In[11]:=11
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-0q09i
Out[11]=11
In[12]:=12
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-0curas
Out[12]=12
In[13]:=13
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-wlr4s0
Out[13]=13

Neat Examples  (1)Surprising or curious use cases

Compute the statistic when the null hypothesis is true:

In[1]:=1
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-2qqg3c
In[2]:=2
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-ywy3ty

The test statistic given a particular alternative:

In[3]:=3
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-612aqt
In[4]:=4
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-c5cy2n

Compare the distributions of the test statistics:

In[5]:=5
✖
https://wolfram.com/xid/0b6g2u59e1po1v6m4hx3k2-87eb6q
Out[5]=5
Wolfram Research (2010), MardiaKurtosisTest, Wolfram Language function, https://reference.wolfram.com/language/ref/MardiaKurtosisTest.html.
✖
Wolfram Research (2010), MardiaKurtosisTest, Wolfram Language function, https://reference.wolfram.com/language/ref/MardiaKurtosisTest.html.

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_mardiakurtosistest, author="Wolfram Research", title="{MardiaKurtosisTest}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/MardiaKurtosisTest.html}", note=[Accessed: 29-May-2025 ]}

✖
@misc{reference.wolfram_2025_mardiakurtosistest, author="Wolfram Research", title="{MardiaKurtosisTest}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/MardiaKurtosisTest.html}", note=[Accessed: 29-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_mardiakurtosistest, organization={Wolfram Research}, title={MardiaKurtosisTest}, year={2010}, url={https://reference.wolfram.com/language/ref/MardiaKurtosisTest.html}, note=[Accessed: 29-May-2025 ]}

✖
@online{reference.wolfram_2025_mardiakurtosistest, organization={Wolfram Research}, title={MardiaKurtosisTest}, year={2010}, url={https://reference.wolfram.com/language/ref/MardiaKurtosisTest.html}, note=[Accessed: 29-May-2025 ]}