KuiperTest
KuiperTest[data]
tests whether data is normally distributed using the Kuiper test.
KuiperTest[data,dist]
tests whether data is distributed according to dist using the Kuiper test.
KuiperTest[data,dist,"property"]
returns the value of "property".
Details and Options
- KuiperTest performs the Kuiper 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 Kuiper test assumes that the data came from a continuous distribution.
- The Kuiper test effectively uses a test statistic based on
where 
is the empirical CDF of data and 
is the CDF of dist. - For multivariate tests, the sum of the univariate marginal
-values is used and is assumed to follow a UniformSumDistribution under 
. - KuiperTest[data,dist,"HypothesisTestData"] returns a HypothesisTestData object htd that can be used to extract additional test results and properties using the form htd["property"].
- KuiperTest[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 
-valuesSignificanceLevel 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 
are generated under 
using the fitted distribution. The EmpiricalDistribution from KuiperTest[si,dist,"TestStatistic"] is then used to estimate the 
-value.
Examples
open all close allBasic Examples (4)
Scope (9)
Testing (6)
Perform a Kuiper 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:
Options (3)
Method (3)
Use Monte Carlo-based methods for 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 (2)
A power curve for the Kuiper test:
Visualize the approximate power curve:
Estimate the power of the Kuiper test when the underlying distribution is a UniformDistribution[{-4,4}], the test size is 0.05, and the sample size is 12:
Spatial signs transform bivariate data to points on a unit circle and are often used in tests of location. The null hypothesis can be tested by determining whether the spatial signs are significantly clustered:
Clustering occurs for nonzero mean vectors:
A bivariate test for location using spatial signs and Kuiper's test:
The test does not reject when the mean is close to the null value:
The test rejects the null hypothesis when the mean is not close to the null value:
Properties & Relations (7)
By default, univariate data is compared to a NormalDistribution:
The parameters have been estimated from the data:
Multivariate data is compared to a 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, KuiperTest 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 Kuiper test works with the values only when the input is a TimeSeries:
Possible Issues (3)
The Kuiper test is not intended for discrete distributions:
The test tends to be quite conservative:
Use Monte Carlo methods or PearsonChiSquareTest in these cases:
The Kuiper 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:
Differences may be more apparent with larger numbers of ties:
Text
Wolfram Research (2010), KuiperTest, Wolfram Language function, https://reference.wolfram.com/language/ref/KuiperTest.html.
CMS
Wolfram Language. 2010. "KuiperTest." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/KuiperTest.html.
APA
Wolfram Language. (2010). KuiperTest. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/KuiperTest.html