Package org.apache.commons.statistics.distribution

Class ParetoDistribution

java.lang.Object
org.apache.commons.statistics.distribution.ParetoDistribution
All Implemented Interfaces:
ContinuousDistribution
public final class ParetoDistribution extends Object
Implementation of the Pareto (Type I) distribution.

The probability density function of \( X \) is:

\[ f(x; k, \alpha) = \frac{\alpha k^\alpha}{x^{\alpha + 1}} \]

for \( k > 0 \), \( \alpha > 0 \), and \( x \in [k, \infty) \).

\( k \) is a scale parameter: this is the minimum possible value of \( X \).
\( \alpha \) is a shape parameter: this is the Pareto index.

See Also:

  • Nested Class Summary

    Nested classes/interfaces inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution

    ContinuousDistribution.Sampler

  • Method Summary

    Modifier and Type
    Method
    Description
    ContinuousDistribution.Sampler
    createSampler(org.apache.commons.rng.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()
    Gets the scale parameter of this distribution.
    double
    getShape()
    Gets 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.
    double
    inverseSurvivalProbability(double p)
    Computes the inverse survival probability function of this distribution.
    double
    logDensity(double x)
    Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified point x.
    static ParetoDistribution
    of(double scale, double shape)
    Creates a Pareto distribution.
    double
    probability(double x0, double x1)
    For a random variable X whose values are distributed according to this distribution, this method returns P(x0 < X <= x1).
    double
    survivalProbability(double x)
    For a random variable X whose values are distributed according to this distribution, this method returns P(X > x).

    Methods inherited from class java.lang.Object

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

  • Method Details

    • of

      public static ParetoDistribution of(double scale, double shape)
      Creates a Pareto distribution.
      Parameters:
      scale - Scale parameter (minimum possible value of X).
      shape - Shape parameter (Pareto index).
      Returns:
      the distribution
      Throws:
      IllegalArgumentException - if scale <= 0, scale is infinite, or shape <= 0.

    • getScale

      public double getScale()
      Gets the scale parameter of this distribution. This is the minimum possible value of X.
      Returns:
      the scale parameter.

    • getShape

      public double getShape()
      Gets the shape parameter of this distribution. This is the Pareto index.
      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 parameter \( k \) and shape parameter \( \alpha \), the PDF is:

      \[ f(x; k, \alpha) = \begin{cases} 0 & \text{for } x \lt k \\ \frac{\alpha k^\alpha}{x^{\alpha + 1}} & \text{for } x \ge k \end{cases} \]

      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 parameter \( k \) and shape parameter \( \alpha \), the CDF is:

      \[ F(x; k, \alpha) = \begin{cases} 0 & \text{for } x \le k \\ 1 - \left( \frac{k}{x} \right)^\alpha & \text{for } x \gt k \end{cases} \]

      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.

    • survivalProbability

      public double survivalProbability(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 complementary cumulative distribution function.

      By default, this is defined as 1 - cumulativeProbability(x), but the specific implementation may be more accurate.

      For scale parameter \( k \) and shape parameter \( \alpha \), the survival function is:

      \[ S(x; k, \alpha) = \begin{cases} 1 & \text{for } x \le k \\ \left( \frac{k}{x} \right)^\alpha & \text{for } x \gt k \end{cases} \]

      Parameters:
      x - Point at which the survival function is evaluated.
      Returns:
      the probability that a random variable with this distribution takes a value greater than x.

    • inverseCumulativeProbability

      public double inverseCumulativeProbability(double p)
      Computes the quantile function of this distribution. For a random variable X distributed according to this distribution, the returned value is:

      \[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \le x) \ge p\} & \text{for } 0 \lt p \le 1 \\ \inf \{ x \in \mathbb R : P(X \le x) \gt 0 \} & \text{for } p = 0 \end{cases} \]

      The default implementation returns:

      Specified by:
      inverseCumulativeProbability in interface ContinuousDistribution
      Parameters:
      p - Cumulative probability.
      Returns:
      the smallest p-quantile of this distribution (largest 0-quantile for p = 0).

    • inverseSurvivalProbability

      public double inverseSurvivalProbability(double p)
      Computes the inverse survival probability function of this distribution. For a random variable X distributed according to this distribution, the returned value is:

      \[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \gt x) \le p\} & \text{for } 0 \le p \lt 1 \\ \inf \{ x \in \mathbb R : P(X \gt x) \lt 1 \} & \text{for } p = 1 \end{cases} \]

      By default, this is defined as inverseCumulativeProbability(1 - p), but the specific implementation may be more accurate.

      The default implementation returns:

      Specified by:
      inverseSurvivalProbability in interface ContinuousDistribution
      Parameters:
      p - Survival probability.
      Returns:
      the smallest (1-p)-quantile of this distribution (largest 0-quantile for p = 1).

    • getMean

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

      For scale parameter \( k \) and shape parameter \( \alpha \), the mean is:

      \[ \mathbb{E}[X] = \begin{cases} \infty & \text{for } \alpha \le 1 \\ \frac{k \alpha}{(\alpha-1)} & \text{for } \alpha \gt 1 \end{cases} \]

      Returns:
      the mean.

    • getVariance

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

      For scale parameter \( k \) and shape parameter \( \alpha \), the variance is:

      \[ \operatorname{var}[X] = \begin{cases} \infty & \text{for } \alpha \le 2 \\ \frac{k^2 \alpha}{(\alpha-1)^2 (\alpha-2)} & \text{for } \alpha \gt 2 \end{cases} \]

      Returns:
      the variance.

    • getSupportLowerBound

      public double getSupportLowerBound()
      Gets the lower bound of the support. It must return the same value as inverseCumulativeProbability(0), i.e. \( \inf \{ x \in \mathbb R : P(X \le x) \gt 0 \} \).

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

      Returns:
      scale.

    • getSupportUpperBound

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

      The upper bound of the support is always positive infinity.

      Returns:
      positive infinity.

    • createSampler

      public ContinuousDistribution.Sampler createSampler(org.apache.commons.rng.UniformRandomProvider rng)
      Creates a sampler.
      Specified by:
      createSampler in interface ContinuousDistribution
      Parameters:
      rng - Generator of uniformly distributed numbers.
      Returns:
      a sampler that produces random numbers according this distribution.

    • probability

      public double probability(double x0, double x1)
      For a random variable X whose values are distributed according to this distribution, this method returns P(x0 < X <= x1). The default implementation uses the identity P(x0 < X <= x1) = P(X <= x1) - P(X <= x0)
      Specified by:
      probability in interface ContinuousDistribution
      Parameters:
      x0 - Lower bound (exclusive).
      x1 - Upper bound (inclusive).
      Returns:
      the probability that a random variable with this distribution takes a value between x0 and x1, excluding the lower and including the upper endpoint.