Documentation of 'umontreal.iro.lecuyer.probdist.CramerVonMisesDist' Java class
CramerVonMisesDist
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umontreal.iro.lecuyer.probdist

Class CramerVonMisesDist

  • All Implemented Interfaces:
    Distribution

    public class CramerVonMisesDist
extends ContinuousDistribution
    Extends the class ContinuousDistribution for the Cramér-von Mises distribution (see). Given a sample of n independent uniforms Ui over [0, 1], the Cramér-von Mises statistic Wn2 is defined by

    Wn2 = 1/12n + ∑j=1n(U(j) - (j-0.5)/n)2,

    where the U(j) are the Ui sorted in increasing order. The distribution function (the cumulative probabilities) is defined as Fn(x) = P[Wn2 <= x].

    • Constructor Summary

      Constructors 
      Constructor and Description
      CramerVonMisesDist(int n)
      Constructs a Cramér-von Mises distribution for a sample of size n.

    • Method Summary

      All Methods Static Methods Instance Methods Concrete Methods 
      Modifier and Type Method and Description
      double barF(double x)
      Returns the complementary distribution function.
      static double barF(int n, double x)
      Computes the complementary distribution function bar(F)n(x) with parameter n.
      double cdf(double x)
      Returns the distribution function F(x).
      static double cdf(int n, double x)
      Computes the Cramér-von Mises distribution function with parameter n.
      double density(double x)
      Returns f (x), the density evaluated at x.
      static double density(int n, double x)
      Computes the density function for a Cramér-von Mises distribution with parameter n.
      double getMean()
      Returns the mean.
      static double getMean(int n)
      Returns the mean of the distribution with parameter n.
      int getN()
      Returns the parameter n of this object.
      double[] getParams()
      Return an array containing the parameter n of this object.
      double getStandardDeviation()
      Returns the standard deviation.
      static double getStandardDeviation(int n)
      Returns the standard deviation of the distribution with parameter n.
      double getVariance()
      Returns the variance.
      static double getVariance(int n)
      Returns the variance of the distribution with parameter n.
      double inverseF(double u)
      Returns the inverse distribution function x = F-1(u).
      static double inverseF(int n, double u)
      Computes x = Fn-1(u), where Fn is the Cramér-von Mises distribution with parameter n.
      void setN(int n)
      Sets the parameter n of this object.
      java.lang.String toString() 

      • Methods inherited from class java.lang.Object

        equals, getClass, hashCode, notify, notifyAll, wait, wait, wait

    • Constructor Detail

      • CramerVonMisesDist

        public CramerVonMisesDist(int n)
        Constructs a Cramér-von Mises distribution for a sample of size n.

    • Method Detail

      • density

        public double density(double x)
        Description copied from class: ContinuousDistribution
        Returns f (x), the density evaluated at x.
        Specified by:
        density in class ContinuousDistribution
        Parameters:
        x - value at which the density is evaluated
        Returns:
        density function evaluated at x

      • cdf

        public double cdf(double x)
        Description copied from interface: Distribution
        Returns the distribution function F(x).
        Parameters:
        x - value at which the distribution function is evaluated
        Returns:
        distribution function evaluated at x

      • barF

        public double barF(double x)
        Description copied from class: ContinuousDistribution
        Returns the complementary distribution function. The default implementation computes bar(F)(x) = 1 - F(x).
        Specified by:
        barF in interface Distribution
        Overrides:
        barF in class ContinuousDistribution
        Parameters:
        x - value at which the complementary distribution function is evaluated
        Returns:
        complementary distribution function evaluated at x

      • inverseF

        public double inverseF(double u)
        Description copied from class: ContinuousDistribution
        Returns the inverse distribution function x = F-1(u). Restrictions: u∈[0, 1].
        Specified by:
        inverseF in interface Distribution
        Overrides:
        inverseF in class ContinuousDistribution
        Parameters:
        u - value at which the inverse distribution function is evaluated
        Returns:
        the inverse distribution function evaluated at u

      • density

        public static double density(int n,
                             double x)
        Computes the density function for a Cramér-von Mises distribution with parameter n.

      • cdf

        public static double cdf(int n,
                         double x)
        Computes the Cramér-von Mises distribution function with parameter n. Returns an approximation of P[Wn2 <= x], where Wn2 is the Cramér von Mises statistic (see). The approximation is based on the distribution function of W2 = limn -> ∞Wn2, which has the following series expansion derived by Anderson and Darling:

                P(W2 <= x)  =  $\displaystyle {\frac{1}{{\pi\sqrt x}}}$∑j=0∞(- 1)j$\displaystyle \binom{-1/2}{j} $(4j+1)1/2    exp{ - $\displaystyle {\frac{{(4j+1)^2}}{{16 x}}}$}K1/4([tex2html_wrap_indisplay246]),

        where Kν is the modified Bessel function of the second kind. To correct for the deviation between P(Wn2 <= x) and P(W2 <= x), we add a correction in 1/n, obtained empirically by simulation. For n = 10, 20, 40, the error is less than 0.002, 0.001, and 0.0005, respectively, while for n >= 100 it is less than 0.0005. For n -> ∞, we estimate that the method returns at least 6 decimal digits of precision. For n = 1, the method uses the exact distribution: P(W12 <= x) = 2(x - 1/12)1/2 for 1/12 <= x <= 1/3.

      • barF

        public static double barF(int n,
                          double x)
        Computes the complementary distribution function bar(F)n(x) with parameter n.

      • inverseF

        public static double inverseF(int n,
                              double u)
        Computes x = Fn-1(u), where Fn is the Cramér-von Mises distribution with parameter n.

      • getMean

        public static double getMean(int n)
        Returns the mean of the distribution with parameter n.
        Returns:
        the mean

      • getVariance

        public static double getVariance(int n)
        Returns the variance of the distribution with parameter n.
        Returns:
        variance

      • getStandardDeviation

        public static double getStandardDeviation(int n)
        Returns the standard deviation of the distribution with parameter n.
        Returns:
        the standard deviation

      • getN

        public int getN()
        Returns the parameter n of this object.

      • setN

        public void setN(int n)
        Sets the parameter n of this object.

      • getParams

        public double[] getParams()
        Return an array containing the parameter n of this object.

      • toString

        public java.lang.String toString()
        Overrides:
        toString in class java.lang.Object
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