KolmogorovSmirnovTest Class 
Namespace: Accord.Statistics.Testing
[SerializableAttribute] public class KolmogorovSmirnovTest : HypothesisTest<KolmogorovSmirnovDistribution>, IHypothesisTest<KolmogorovSmirnovDistribution>, IHypothesisTest
The KolmogorovSmirnovTest type exposes the following members.
Name  Description  

KolmogorovSmirnovTest(Double, IDistributionDouble) 
Creates a new OneSample Kolmogorov test.
 
KolmogorovSmirnovTest(Double, IDistributionDouble, KolmogorovSmirnovTestHypothesis) 
Creates a new OneSample Kolmogorov test.

Name  Description  

CriticalValue 
Gets the critical value for the current significance level.
(Inherited from HypothesisTestTDistribution.)  
EmpiricalDistribution 
Gets the empirical distribution measured from the sample.
 
Hypothesis 
Gets the alternative hypothesis under test. If the test is
Significant, the null hypothesis can be rejected
in favor of this alternative hypothesis.
 
PValue 
Gets the Pvalue associated with this test.
(Inherited from HypothesisTestTDistribution.)  
Significant 
Gets whether the null hypothesis should be rejected.
(Inherited from HypothesisTestTDistribution.)  
Size 
Gets the significance level for the
test. Default value is 0.05 (5%).
(Inherited from HypothesisTestTDistribution.)  
Statistic 
Gets the test statistic.
(Inherited from HypothesisTestTDistribution.)  
StatisticDistribution 
Gets the distribution associated
with the test statistic.
(Inherited from HypothesisTestTDistribution.)  
Tail 
Gets the test type.
(Inherited from HypothesisTestTDistribution.)  
TheoreticalDistribution 
Gets the theoretical, hypothesized distribution for the samples,
which should have been stated before any measurements.

Name  Description  

Equals  Determines whether the specified object is equal to the current object. (Inherited from Object.)  
Finalize  Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)  
GetHashCode  Serves as the default hash function. (Inherited from Object.)  
GetStatistic 
Gets the appropriate KolmogorovSminorv D statistic for the samples and target distribution.
 
GetType  Gets the Type of the current instance. (Inherited from Object.)  
MemberwiseClone  Creates a shallow copy of the current Object. (Inherited from Object.)  
OneSideLower 
Gets the onesided "Dn" KolmogorovSminorv statistic for the samples and target distribution.
 
OneSideUpper 
Gets the onesided "Dn+" KolmogorovSminorv statistic for the samples and target distribution.
 
OnSizeChanged 
Called whenever the test significance level changes.
(Inherited from HypothesisTestTDistribution.)  
PValueToStatistic 
Converts a given pvalue to a test statistic.
(Overrides HypothesisTestTDistributionPValueToStatistic(Double).)  
StatisticToPValue 
Converts a given test statistic to a pvalue.
(Overrides HypothesisTestTDistributionStatisticToPValue(Double).)  
ToString 
Converts the numeric PValue of this test to its equivalent string representation.
(Inherited from HypothesisTestTDistribution.)  
ToString(String, IFormatProvider) 
Converts the numeric PValue of this test to its equivalent string representation.
(Inherited from HypothesisTestTDistribution.)  
TwoSide 
Gets the twosided "Dn" KolmogorovSminorv statistic for the samples and target distribution.

Name  Description  

HasMethod 
Checks whether an object implements a method with the given name.
(Defined by ExtensionMethods.)  
IsEqual 
Compares two objects for equality, performing an elementwise
comparison if the elements are vectors or matrices.
(Defined by Matrix.)  
To(Type)  Overloaded.
Converts an object into another type, irrespective of whether
the conversion can be done at compile time or not. This can be
used to convert generic types to numeric types during runtime.
(Defined by ExtensionMethods.)  
ToT  Overloaded.
Converts an object into another type, irrespective of whether
the conversion can be done at compile time or not. This can be
used to convert generic types to numeric types during runtime.
(Defined by ExtensionMethods.) 
The KolmogorovSmirnov test tries to determine if a sample differs significantly from an hypothesized theoretical probability distribution. The KolmogorovSmirnov test has an interesting advantage in which it does not requires any assumptions about the data. The distribution of the KS test statistic does not depend on which distribution is being tested.
The KS test has also the advantage of being an exact test (other tests, such as the chisquare goodnessoffit test depends on an adequate sample size). One disadvantage is that it requires a fully defined distribution which should not have been estimated from the data. If the parameters of the theoretical distribution have been estimated from the data, the critical region of the KS test will be no longer valid.
This class uses an efficient and highaccuracy algorithm based on work by Richard Simard (2010). Please see KolmogorovSmirnovDistribution for more details.
References:
In this first example, suppose we got a new sample, and we would like to test whether this sample has been originated from a uniform continuous distribution.
double[] sample = { 0.621, 0.503, 0.203, 0.477, 0.710, 0.581, 0.329, 0.480, 0.554, 0.382 }; // First, we create the distribution we would like to test against: // var distribution = UniformContinuousDistribution.Standard; // Now we can define our hypothesis. The null hypothesis is that the sample // comes from a standard uniform distribution, while the alternate is that // the sample is not from a standard uniform distribution. // var kstest = new KolmogorovSmirnovTest(sample, distribution); double statistic = kstest.Statistic; // 0.29 double pvalue = kstest.PValue; // 0.3067 bool significant = kstest.Significant; // false
Since the null hypothesis could not be rejected, then the sample can perhaps be from a uniform distribution. However, please note that this doesn't means that the sample *is* from the uniform, it only means that we could not rule out the possibility.
Before we could not rule out the possibility that the sample came from a uniform distribution, which means the sample was not very far from uniform. This would be an indicative that it would be far from what would be expected from a Normal distribution:
// First, we create the distribution we would like to test against: // NormalDistribution distribution = NormalDistribution.Standard; // Now we can define our hypothesis. The null hypothesis is that the sample // comes from a standard Normal distribution, while the alternate is that // the sample is not from a standard Normal distribution. // var kstest = new KolmogorovSmirnovTest(sample, distribution); double statistic = kstest.Statistic; // 0.580432 double pvalue = kstest.PValue; // 0.000999 bool significant = kstest.Significant; // true
Since the test says that the null hypothesis should be rejected, then this can be regarded as a strong indicative that the sample does not comes from a Normal distribution, just as we expected.