Package org.apache.commons.statistics.distribution

Class ParetoDistribution

  • All Implemented Interfaces:
    ContinuousDistribution
    public class ParetoDistribution
extends java.lang.Object
    Implementation of the Pareto distribution.

    Parameters: The probability distribution function of X is given by (for x >= k): 

      α * k^α / x^(α + 1)
    • k is the scale parameter: this is the minimum possible value of X,
    • α is the shape parameter: this is the Pareto index

    • Constructor Summary

      Constructors 
      Constructor Description
      ParetoDistribution​(double scale, double shape)
      Creates a Pareto distribution.

    • Method Summary

      Modifier and Type Method Description
      ContinuousDistribution.Sampler createSampler​(UniformRandomProvider rng)
      Creates a sampler.
      double cumulativeProbability​(double x)
      For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x).
      double density​(double x)
      Returns the probability density function (PDF) of this distribution evaluated at the specified point x.
      double getMean()
      Gets the mean of this distribution.
      double getScale()
      Returns the scale parameter of this distribution.
      double getShape()
      Returns the shape parameter of this distribution.
      double getSupportLowerBound()
      Gets the lower bound of the support.
      double getSupportUpperBound()
      Gets the upper bound of the support.
      double getVariance()
      Gets the variance of this distribution.
      double inverseCumulativeProbability​(double p)
      Computes the quantile function of this distribution.
      boolean isSupportConnected()
      Indicates whether the support is connected, i.e.
      double logDensity​(double x)
      Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified point x.
      static double[] sample​(int n, ContinuousDistribution.Sampler sampler)
      Utility function for allocating an array and filling it with n samples generated by the given sampler.

      • Methods inherited from class java.lang.Object

        clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

    • Constructor Detail

      • ParetoDistribution

        public ParetoDistribution​(double scale,
                          double shape)
        Creates a Pareto distribution.
        Parameters:
        scale - Scale parameter of this distribution.
        shape - Shape parameter of this distribution.
        Throws:
        java.lang.IllegalArgumentException - if scale <= 0 or shape <= 0.

    • Method Detail

      • getScale

        public double getScale()
        Returns the scale parameter of this distribution.
        Returns:
        the scale parameter

      • getShape

        public double getShape()
        Returns the shape parameter of this distribution.
        Returns:
        the shape parameter

      • density

        public double density​(double x)
        Returns the probability density function (PDF) of this distribution evaluated at the specified point x. In general, the PDF is the derivative of the CDF. If the derivative does not exist at x, then an appropriate replacement should be returned, e.g. Double.POSITIVE_INFINITY, Double.NaN, or the limit inferior or limit superior of the difference quotient.

        For scale k, and shape α of this distribution, the PDF is given by

        • 0 if x < k,
        • α * k^α / x^(α + 1) otherwise.
        Parameters:
        x - Point at which the PDF is evaluated.
        Returns:
        the value of the probability density function at x.

      • logDensity

        public double logDensity​(double x)
        Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified point x. See documentation of density(double) for computation details.
        Parameters:
        x - Point at which the PDF is evaluated.
        Returns:
        the logarithm of the value of the probability density function at x.

      • cumulativeProbability

        public double cumulativeProbability​(double x)
        For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x). In other words, this method represents the (cumulative) distribution function (CDF) for this distribution.

        For scale k, and shape α of this distribution, the CDF is given by

        • 0 if x < k,
        • 1 - (k / x)^α otherwise.
        Parameters:
        x - Point at which the CDF is evaluated.
        Returns:
        the probability that a random variable with this distribution takes a value less than or equal to x.

      • getMean

        public double getMean()
        Gets the mean of this distribution.

        For scale k and shape α, the mean is given by

        • ∞ if α <= 1,
        • α * k / (α - 1) otherwise.
        Returns:
        the mean, or Double.NaN if it is not defined.

      • getVariance

        public double getVariance()
        Gets the variance of this distribution.

        For scale k and shape α, the variance is given by

        • ∞ if 1 < α <= 2,
        • k^2 * α / ((α - 1)^2 * (α - 2)) otherwise.
        Returns:
        the variance, or Double.NaN if it is not defined.

      • getSupportLowerBound

        public double getSupportLowerBound()
        Gets the lower bound of the support. It must return the same value as inverseCumulativeProbability(0), i.e. inf {x in R | P(X <= x) > 0}.

        The lower bound of the support is equal to the scale parameter k.

        Returns:
        lower bound of the support

      • getSupportUpperBound

        public double getSupportUpperBound()
        Gets the upper bound of the support. It must return the same value as inverseCumulativeProbability(1), i.e. inf {x in R | P(X <= x) = 1}.

        The upper bound of the support is always positive infinity no matter the parameters.

        Returns:
        upper bound of the support (always Double.POSITIVE_INFINITY)

      • isSupportConnected

        public boolean isSupportConnected()
        Indicates whether the support is connected, i.e. whether all values between the lower and upper bound of the support are included in the support.

        The support of this distribution is connected.

        Returns:
        true

      • sample

        public static double[] sample​(int n,
                              ContinuousDistribution.Sampler sampler)
        Utility function for allocating an array and filling it with n samples generated by the given sampler.
        Parameters:
        n - Number of samples.
        sampler - Sampler.
        Returns:
        an array of size n.