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- is the set of values.
- This function gives the test statistic of the K-S test.
- K-S test is indicating the Kolmogorov-Smirnov test.
- It is one of the non parametric test.
- This test is the equality of continuous one dimensional probability distribution.
- It can be used to compare sample with a reference probability distribution or to compare two samples.
- This test statistic measures a distance between the empirical distribution function of the sample and the cumulative distribution function of the reference distribution, or between the empirical distribution functions of two samples.
- The two-sample KS test is one of the most useful and general nonparametric methods for comparing two samples.
- It is sensitive to differences in both location and shape of the empirical cumulative distribution functions of the two samples.
- This test can be modified to serve as a goodness of fit test.
- The assumption of the KS test is:
- Null Hypothesis(H0):The sampled population is normally distributed.
- Alternative hypothesis(Ha):The sampled population is not normally distributed.
- The Kolmogorov-Smirnov test to compare a data set to a given theoretical distribution is as follows:
- 1.Data set sorted into increasing order and denoted as , where i=1,...,n.
- 2.Smallest empirical estimate of fraction of points falling below , and computed as for i=1,...,n.
- 3.Largest empirical estimate of fraction of points falling below and computed as for i=1,...,n.
- 4.Theoretical estimate of fraction of points falling below and computed as , where F(x) is the theoretical distribution function being tested.
- 5.Find the absolute value of difference of Smallest and largest empirical value with the theoretical estimation of points.
- This is a measure of "error" for this data point.
- 6.From the largest error, we can compute the test statistic.
- The Kolmogorov-Smirnov test statistic for the cumulative distribution F(x) is:where is the supremum of the set of distances.
- is the empirical distribution function for n,with the observations is defined as:where is the indicator function, equal to 1 if and equal to 0 otherwise.
- Using this function we can identify the following deatils:
Are the data from the Normal distribution or Weibull distribution or Exponential distribution or a logistic distribution.