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MardiaKurtosisTest

MardiaKurtosisTest[data]
tests whether data follows a MultinormalDistribution using the Mardia kurtosis test.
MardiaKurtosisTest
returns the value of .
  • By default a probability value or -value is returned.
  • A small -value suggests that it is unlikely that the data came from dist.
  • The data can be univariate or multivariate .
  • The Mardia kurtosis test effectively compares a multivariate measure of kurtosis for data to a MultinormalDistribution.
  • Properties related to the reporting of test results include:
"DegreesOfFreedom"the degrees of freedom used in a test
"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:
MethodAutomaticthe 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 and properties is controlled by the SignificanceLevel option. By default is set to .
  • 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
Perform a test for multivariate normality:
Extract the test statistic from the Mardia kurtosis test:
Obtain a formatted test table:
Perform a test for multivariate normality:
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Extract the test statistic from the Mardia kurtosis test:
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Obtain a formatted test table:
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Perform a Mardia kurtosis test for multivariate normality:
The -value for the normal data is large compared to the -value for the non-normal data:
Create a HypothesisTestData object for repeated property extraction:
The properties available for extraction:
Tabulate the results of the Mardia kurtosis test:
The full test table:
A -value table:
The test statistic:
Retrieve the entries from a Mardia kurtosis test table for custom reporting:
Report test conclusions using and :
The conclusion may differ at a different significance level:
Use Monte Carlo-based methods or 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:
Set the significance level used for and :
By default is used:
A power curve for the Mardia kurtosis test:
Visualize the approximate power curve:
Estimate the power of the Mardia kurtosis test when the underlying distribution is a MultivariateTDistribution, the test size is , and the sample size is 27:
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:
The multivariate kurtosis of the two species is similar to a multivariate normal distribution:
The multivariate skewness should also be checked to confirm normality:
The data appears normal so TTest is valid:
The multivariate test statistic:
The univariate test statistic:
The multivariate test statistic has an asymptotic NormalDistribution:
The asymptotic -value can be very inaccurate for small samples:
For comparison, the Monte Carlo -value is much closer to the small-sample value:
Mardia's kurtosis test can only detect departures from normality in kurtosis:
The data is clearly not normally distributed:
Decisions should be based on MardiaSkewnessTest and MardiaKurtosisTest:
If the covariance matrix of the data is not positive definite the test will fail:
The number of data points must be greater than the dimension of the data:
The distribution of the Mardia kurtosis test statistic:
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