JMSLTM Numerical Library 6.1

com.imsl.stat
Class KolmogorovTwoSample

java.lang.Object
  extended by com.imsl.stat.KolmogorovTwoSample
All Implemented Interfaces:
Serializable

public class KolmogorovTwoSample
extends Object
implements Serializable

Performs a Kolmogorov-Smirnov two-sample test.

Class KolmogorovTwoSample computes Kolmogorov-Smirnov two-sample test statistics for testing that two continuous cumulative distribution functions (CDF's) are identical based upon two random samples. One- or two-sided alternatives are allowed. Exact p-values are computed for the two-sided test when nm le 104, where n is the number of non-missing X observations and m the number of non-missing Y observation.

Let F_n(x) denote the empirical CDF in the X sample, let G_m(y) denote the empirical CDF in the Y sample and let the corresponding population distribution functions be denoted by F(x) and G(y), respectively. Then, the hypotheses tested by KolmogorovTwoSample are as follows:

begin{array}{ll}
         H_0:~ F(x) = G(x)   & H_1:~F(x) ne G(x) \
         H_0:~ F(x) ge G(x) & H_1:~F(x) lt G(x) \
         H_0:~ F(x) le G(x) & H_1:~F(x) gt G(x)
     end{array}

The test statistics are given as follows:

begin{array}{rl}
      D_{mn}     & = max(D_{mn}^{+}, D_{mn}^{-}) \
      D_{mn}^{+} & = max_x(F_n(x)-G_m(x)) \
      D_{mn}^{-} & = max_x(G_m(x)-F_n(x))
 end{array}

Asymptotically, the distribution of the statistic

Z = D_{mn} sqrt{frac{m+n}{mn}}

converges to a distribution given by Smirnov (1939).

Exact probabilities for the two-sided test are computed when nm le 104, according to an algorithm given by Kim and Jennrich (1973). When nm gt 104, the very good approximations given by Kim and Jennrich are used to obtain the two-sided p-values. The one-sided probability is taken as one half the two-sided probability. This is a very good approximation when the p-value is small (say, less than 0.10) and not very good for large p-values.

See Also:
Example, Serialized Form

Constructor Summary
KolmogorovTwoSample(double[] x, double[] y)
          Constructs a two sample Kolmogorov-Smirnov goodness-of-fit test.
 
Method Summary
 double getMaximumDifference()
          Returns D^{+}, the maximum difference between the theoretical and empirical CDF's.
 double getMinimumDifference()
          Returns D^{-}, the minimum difference between the theoretical and empirical CDF's.
 int getNumberMissingX()
          Returns the number of missing values in the x sample.
 int getNumberMissingY()
          Returns the number of missing values in the y sample.
 double getOneSidedPValue()
          Probability of the statistic exceeding D under the null hypothesis of equality and against the one-sided alternative.
 double getTestStatistic()
          Returns D = max(D^{+}, D^{-}).
 double getTwoSidedPValue()
          Probability of the statistic exceeding D under the null hypothesis of equality and against the two-sided alternative.
 double getZ()
          Returns the normalized D statistic without the continuity correction applied.
 
Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
 

Constructor Detail

KolmogorovTwoSample

public KolmogorovTwoSample(double[] x,
                           double[] y)
Constructs a two sample Kolmogorov-Smirnov goodness-of-fit test.

Parameters:
x - is an array containing the observations from the first sample.
y - is an array containing the observations from the second sample.
Method Detail

getMaximumDifference

public double getMaximumDifference()
Returns D^{+}, the maximum difference between the theoretical and empirical CDF's.

Returns:
The value D^{+}.

getMinimumDifference

public double getMinimumDifference()
Returns D^{-}, the minimum difference between the theoretical and empirical CDF's.

Returns:
The value D^{-}.

getNumberMissingX

public int getNumberMissingX()
Returns the number of missing values in the x sample.

Returns:
The number of missing values in x.

getNumberMissingY

public int getNumberMissingY()
Returns the number of missing values in the y sample.

Returns:
The number of missing values in y.

getOneSidedPValue

public double getOneSidedPValue()
Probability of the statistic exceeding D under the null hypothesis of equality and against the one-sided alternative. An exact probability is computed if the number of observation is less than or equal to 80, otherwise an approximate probability is computed.

Returns:
the one-sided probability.

getTestStatistic

public double getTestStatistic()
Returns D = max(D^{+}, D^{-}).

Returns:
The value D.

getTwoSidedPValue

public double getTwoSidedPValue()
Probability of the statistic exceeding D under the null hypothesis of equality and against the two-sided alternative. This probability is twice the probability, p_1, reported by getOneSidedPValue, (or 1.0 if p_1 ge 1/2. This approximation is nearly exact when p_1 lt 0.1.

Returns:
the two-sided probability.

getZ

public double getZ()
Returns the normalized D statistic without the continuity correction applied.

Returns:
the value Z

JMSLTM Numerical Library 6.1

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Built July 30 2010.