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

# CramerVonMisesTest

 CramerVonMisesTest[data] tests whether data is normally distributed using the Cramér-von Mises test. CramerVonMisesTest tests whether data is distributed according to dist using the Cramér-von Mises test. CramerVonMisesTest returns the value of .
• CramerVonMisesTest performs the Cramér-von Mises goodness-of-fit test with null hypothesis that data was drawn from a population with distribution dist 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 Cramér-von Mises test assumes that the data came from a continuous distribution.
• The Cramér-von Mises test effectively uses a test statistic based on the expectation value of where is the empirical CDF of data and is the CDF of dist.
• For univariate data, the test statistic is given by .
• 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:
 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 .
• With the setting Method, datasets of the same length as the input are generated under using the fitted distribution. The empirical distribution from CramerVonMisesTest is then used to estimate the -value.
Perform a Cramér-von Mises test for normality:
Confirm the result using QuantilePlot:
Test the fit of some data to a particular distribution:
Compare the distributions of two datasets:
Perform a Cramér-von Mises test for normality:
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Confirm the result using QuantilePlot:
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Test the fit of some data to a particular distribution:
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Compare the distributions of two datasets:
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 Scope   (9)
Perform a Cramér-von Mises test for normality:
The -value for the normal data is large compared to the -value for the non-normal data:
Test the goodness of fit to a particular distribution:
Compare the distributions of two datasets:
The two datasets do not have the same distribution:
Test for multivariate normality:
Test for goodness of fit to any multivariate distribution:
Create a HypothesisTestData object for repeated property extraction:
The properties available for extraction:
Tabulate the results of the Cramér-von Mises test:
The full test table:
A -value table:
The test statistic:
Retrieve the entries from a Cramér-von Mises test table for custom reporting:
Report test conclusions using and :
The conclusion may differ at a different significance level:
 Options   (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:
 Applications   (3)
A power curve for the Cramér-von Mises test:
Visualize the approximate power curve:
Estimate the power of the Cramér-von Mises test when the underlying distribution is UniformDistribution, the test size is , and the sample size is 32:
Observations generated by a homogeneous Poisson process should exhibit complete spatial randomness, which implies that they were drawn from a uniform distribution. Determine if observations from the following images would be modeled well by a homogeneous Poisson process:
Find the centers of each point and rescale on :
The points in the first image would be modeled well by a homogeneous Poisson process:
A model for the second group should account for dependence:
Find the parameters for distributions that minimize the Cramér-von Mises test statistic:
Compare the results to FindDistributionParameters:
By default, univariate data is compared to NormalDistribution:
The parameters have been estimated from the data:
Multivariate data is compared to MultinormalDistribution by default:
The parameters of the test distribution are estimated from the data if not specified:
Specified parameters are not estimated:
Maximum likelihood estimates are used for unspecified parameters of the test distribution:
If the parameters are unknown, CramerVonMisesTest applies a correction when possible:
The parameters are estimated but no correction is applied:
The fitted distribution is the same as before and the -value is corrected:
Independent marginal densities are assumed in tests for multivariate goodness of fit:
The test statistic is identical when independence is assumed:
The Cramér-von Mises test is not intended for discrete distributions:
The Cramér-von Mises test is not valid for some distributions when parameters have been estimated from the data:
Provide parameter values if they are known:
Alternatively, use Monte Carlo methods to approximate the -value:
The Cramér-von Mises test must have sample sizes of at least 7 for valid -values:
Use Monte Carlo methods to arrive at a valid -value:
The distribution of the Cramér-von Mises test statistic:
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