The KolmogorovSmirnov Test

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The Kolmogorov-Smirnov Test

• Vasileios Hatzivassiloglou
• University of Texas at Dallas

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Kolmogorov-Smirnov test

• A fully non-parametric test for comparing two
  distributions
• Does not depend on approximations for the
  distribution

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Empirical distribution function

• For a random variable X and a sample x1, x2,
  ..., xn the empirical distribution function of X
  is defined as
• where I(condition) is the indicator function,
  i.e., 1 if the condition is true and 0 otherwise

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Example data

• FX is an estimate of
• the cumulative probability function of X
• Consider the following example data
• 1.26, 0.34, 0.70, 1.75, 50.57, 1.55, 0.08, 0.42,
  0.50, 3.20, 0.15, 0.49, 0.95, 0.24, 1.37, 0.17,
  6.98, 0.10, 0.94, 0.38
• n 20
• Is this data normal?

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Examining the data

• Sorted data
• 0.08, 0.10, 0.15, 0.17, 0.24, 0.34, 0.38, 0.42,
  0.49, 0.50, 0.70, 0.94, 0.95, 1.26, 1.37, 1.55,
  1.75, 3.20, 6.98, 50.57
• Mean 3.61, Standard deviation 11.2

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Examining the data for normality

• For normal data,
• 15 should be below one s.d. from the mean
  (3.61-11.2 -7.59)
• none of the samples are even negative
• about 2 should be above two standard deviations
  from the mean (3.61211.226.01)
• here we have one in 20 samples way beyond that
  value (50.57)

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Example empirical distribution
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Log transformation
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The Kolmogorov-Smirnov test

• Given two cumulative probability functions FX and
  FY, the test statistics are
• Usually the value DmaxD, D- is used (although
  its distribution is harder to study than either
  D or D-)

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Comparing distributions
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Advantages of Kolmogorov-Smirnov

• It is non-parametric and hence robust
• It does not rely on the means location only
  (like the t-test)
• It works for non-normal data (the t-test can fail
  if the data is too far from normal)
• It is not sensitive to scaling
• It is more powerful than ?2
• However, it is less sensitive than t if the data
  is indeed normal