MardiaKurtosisTest

MardiaKurtosisTest[data]

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 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 "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:

	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-> "MonteCarlo", datasets of the same length as the input are generated under using the fitted distribution. The EmpiricalDistribution from "MonteCarlo" MardiaKurtosisTest[s_i,"TestStatistic"] is then used to estimate the -value.

Examples

open allclose all

Basic Examples (3)

Perform a test for multivariate normality:

Extract the test statistic from the Mardia kurtosis test:

Obtain a formatted test table:

Scope (5)

Testing (2)

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:

Reporting (3)

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 "ShortTestConclusion" and "TestConclusion":

The conclusion may differ at a different significance level:

Options (4)

Method (3)

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:

SignificanceLevel (1)

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

By default, 0.05 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 0.05, 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 (5)

The multivariate test statistic:

The univariate test statistic:

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

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:

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

Possible Issues (1)

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:

Neat Examples (1)

Compute the statistic when the null hypothesis is true:

The test statistic given a particular alternative:

Compare the distributions of the test statistics:

Top

MardiaKurtosisTest

Details and Options

Examples

Basic Examples (3)