# CramerVonMisesTest

CramerVonMisesTest[data]

tests whether data is normally distributed using the Cramérvon Mises test.

CramerVonMisesTest[data,dist]

tests whether data is distributed according to dist using the Cramérvon Mises test.

CramerVonMisesTest[data,dist,"property"]

returns the value of "property".

# Details and Options   • CramerVonMisesTest performs the Cramérvon 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 {x1,x2,} or multivariate {{x1,y1,},{x2,y2,},}.
• The Cramérvon Mises test assumes that the data came from a continuous distribution.
• The Cramérvon 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 sum of the univariate marginal -values is used and is assumed to follow a UniformSumDistribution under .
• CramerVonMisesTest[data,dist,"HypothesisTestData"] returns a HypothesisTestData object htd that can be used to extract additional test results and properties using the form htd["property"].
• CramerVonMisesTest[data,dist,"property"] can be used to directly give the value of "property".
• Properties related to the reporting of test results include:
•  "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.
• With the setting Method->"MonteCarlo", datasets of the same length as the input si are generated under using the fitted distribution. The empirical distribution from CramerVonMisesTest[si,dist,"TestStatistic"] is then used to estimate the -value.

# Examples

open allclose all

## Basic Examples(3)

Perform a Cramérvon Mises test for normality:

 In:= In:= Out= Confirm the result using QuantilePlot:

 In:= Out= Test the fit of some data to a particular distribution:

 In:= In:= Out= In:= Out= Compare the distributions of two datasets:

 In:= In:= In:= Out= In:= Out= 