Package org.hipparchus.distribution.continuous

Class ParetoDistribution

java.lang.Object
org.hipparchus.distribution.continuous.AbstractRealDistribution
org.hipparchus.distribution.continuous.ParetoDistribution
All Implemented Interfaces:
Serializable, RealDistribution
public class ParetoDistribution extends AbstractRealDistribution
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
See Also:

  • Constructor Details

    • ParetoDistribution

      public ParetoDistribution()
      Create a Pareto distribution with a scale of 1 and a shape of 1.

    • ParetoDistribution

      public ParetoDistribution(double scale, double shape) throws MathIllegalArgumentException
      Create a Pareto distribution using the specified scale and shape.
      Parameters:
      scale - the scale parameter of this distribution
      shape - the shape parameter of this distribution
      Throws:
      MathIllegalArgumentException - if scale <= 0 or shape <= 0.

    • ParetoDistribution

      public ParetoDistribution(double scale, double shape, double inverseCumAccuracy) throws MathIllegalArgumentException
      Creates a Pareto distribution.
      Parameters:
      scale - Scale parameter of this distribution.
      shape - Shape parameter of this distribution.
      inverseCumAccuracy - Inverse cumulative probability accuracy.
      Throws:
      MathIllegalArgumentException - if scale <= 0 or shape <= 0.

  • Method Details

    • 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 - the point at which the PDF is evaluated
      Returns:
      the value of the probability density function at point 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. 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. Note that due to the floating point precision and under/overflow issues, this method will for some distributions be more precise and faster than computing the logarithm of RealDistribution.density(double).

      The default implementation simply computes the logarithm of density(x). See documentation of density(double) for computation details.

      Specified by:
      logDensity in interface RealDistribution
      Overrides:
      logDensity in class AbstractRealDistribution
      Parameters:
      x - the point at which the PDF is evaluated
      Returns:
      the logarithm of the value of the probability density function at point 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 - the 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

    • getNumericalMean

      public double getNumericalMean()
      Use this method to get the numerical value of 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

    • getNumericalVariance

      public double getNumericalVariance()
      Use this method to get the numerical value of 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 (possibly Double.POSITIVE_INFINITY as for certain cases in TDistribution) or Double.NaN if it is not defined

    • getSupportLowerBound

      public double getSupportLowerBound()
      Access the lower bound of the support. This method must return the same value as inverseCumulativeProbability(0). In other words, this method must return

      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()
      Access the upper bound of the support. This method must return the same value as inverseCumulativeProbability(1). In other words, this method must return

      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()
      Use this method to get information about 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