BUILT-IN MATHEMATICA SYMBOL

MardiaKurtosisTest

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

MardiaKurtosisTest

returns the value of

.

MORE INFORMATION

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 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.

MardiaKurtosisTest returns a HypothesisTestData object htd that can be used to extract additional test results and properties using the form htd["property"].

MardiaKurtosisTest can be used to directly give the value of .

PearsonChiSquareTest can be used to directly give the value of .

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

With the setting Method, datasets of the same length as the input are generated under using the fitted distribution. The EmpiricalDistribution from MardiaKurtosisTest is then used to estimate the -value.

EXAMPLES

CLOSE ALL

Basic Examples (3)

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:

In[1]:=

In[2]:=

Out[2]=

Extract the test statistic from the Mardia kurtosis test:

In[1]:=

In[2]:=

Out[2]=

Obtain a formatted test table:

In[1]:=

In[2]:=

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:

Properties & Relations (4)

The multivariate test statistic:

The univariate test statistic:

The multivariate test statistic has an asymptotic NormalDistribution

: