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

# 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:
 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 .
• The following methods can be used to compute -values:
 Automatic correct 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:
 Out[2]=

Extract the test statistic from the Mardia kurtosis test:
 Out[2]=

Obtain a formatted test table:
 Out[2]=
 Scope   (5)
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:
 Options   (4)
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:
 Applications   (2)
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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