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JarqueBeraALMTest

JarqueBeraALMTest[data]
tests whether data is normally distributed using the Jarque-Bera ALM test.
JarqueBeraALMTest
returns the value of .
  • JarqueBeraALMTest performs the Jarque-Bera ALM goodness-of-fit test with null hypothesis that data was drawn from a NormalDistribution 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 dist can be any symbolic distribution with numeric and symbolic parameters or a dataset.
  • The data can be univariate or multivariate .
  • The Jarque-Bera ALM test effectively compares the skewness and kurtosis of data to a NormalDistribution.
  • For univariate data the test statistic is given by with , and correction factors for finite sample sizes given by , , and .
  • For multivariate tests, the mean of the univariate marginal test statistics is used. -values are computed via Monte Carlo simulation.
  • Properties related to the reporting of test results include:
"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 .
Perform a Jarque-Bera ALM test for normality:
Perform a test for multivariate normality:
Extract the test statistic from a Jarque-Bera ALM test:
Perform a Jarque-Bera ALM test for normality:
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Perform a test for multivariate normality:
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Extract the test statistic from a Jarque-Bera ALM test:
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Perform a Jarque-Bera ALM test for normality:
The -value for the normal data is large compared to the -value for the non-normal data:
Test for multivariate normality:
Create a HypothesisTestData object for repeated property extraction:
The properties available for extraction:
Tabulate the results of the Jarque-Bera ALM test:
The full test table:
A -value table:
The test statistic:
Retrieve the entries from a Jarque-Bera ALM 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:
A power curve for the Jarque-Bera ALM test:
Visualize the approximate power curve:
Estimate the power of the Jarque-Bera ALM test when the underlying distribution is a CauchyDistribution, the test size is 0.05, and the sample size is 12:
Create a Jarque-Bera ALM test statistic generalized for other distributions:
Finite-sample values for , , and :
A Jarque-Bera ALM test statistic for fitting to a LaplaceDistribution:
Perform the generalized test on some data:
The -values are uniform as expected:
The test is powerful against the alternative of a HyperbolicDistribution of similar mean and variance:
The Adjusted Lagrange Multiplier (ALM) method outperforms the traditional Jarque-Bera test:
The traditional Jarque-Bera test statistic:
The -values are not uniformly distributed:
The Jarque-Bera ALM test is superior for small samples:
The Jarque-Bera ALM test uses finite-sample values for the mean and variance of skewness and kurtosis, not the asymptotic values of 0, 6, 3, and 24 as in the traditional test:
The finite-sample values can be derived using MomentEvaluate and MomentConvert:
The test statistics have the same asymptotic distribution:
The Jarque-Bera ALM test requires sample sizes larger than 9 for -values to be valid:
The distribution of the Jarque-Bera ALM test statistic:
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