KolmogorovSmirnovTest Class

The Kolmogorov-Smirnov test tries to determine if a sample differs significantly from an hypothesized theoretical probability distribution. The Kolmogorov-Smirnov test has an interesting advantage in which it does not requires any assumptions about the data. The distribution of the K-S test statistic does not depend on which distribution is being tested.

The K-S test has also the advantage of being an exact test (other tests, such as the chi-square goodness-of-fit 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of the theoretical distribution have been estimated from the data, the critical region of the K-S test will be no longer valid.

This class uses an efficient and high-accuracy algorithm based on work by Richard Simard (2010). Please see KolmogorovSmirnovDistribution for more details.

References:

• Wikipedia, The Free Encyclopedia. Kolmogorov-Smirnov Test. Available on: http://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test
• NIST/SEMATECH e-Handbook of Statistical Methods. Kolmogorov-Smirnov Goodness-of-Fit Test. Available on: http://www.itl.nist.gov/div898/handbook/eda/section3/eda35g.htm
• Richard Simard, Pierre L’Ecuyer. Computing the Two-Sided Kolmogorov-Smirnov Distribution. Journal of Statistical Software. Volume VV, Issue II. Available on: http://www.iro.umontreal.ca/~lecuyer/myftp/papers/ksdist.pdf

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.

Reference