Package com.imsl.stat

Class KolmogorovTwoSample

java.lang.Object
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 $$ Z = D_{mn} \sqrt{\frac{mn}{m+n}} $$ 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:

  • Constructor Summary

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

  • Method Summary

    Modifier and Type
    Method
    Description
    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 Details

    • 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 Details

    • getTestStatistic

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

    • 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 \(D^{-}\).

    • getZ

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

    • 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.

    • 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.

    • 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.