Class ParetoDistribution

java.lang.Object
org.apache.commons.statistics.distribution.AbstractContinuousDistribution
org.apache.commons.statistics.distribution.ParetoDistribution
All Implemented Interfaces:
ContinuousDistribution

public final class ParetoDistribution extends AbstractContinuousDistribution
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:
  • Field Details

    • MIN_SHAPE_FOR_VARIANCE

      private static final double MIN_SHAPE_FOR_VARIANCE
      The minimum value for the shape parameter when computing when computing the variance.
      See Also:
    • scale

      private final double scale
      The scale parameter of this distribution. Also known as k; the minimum possible value for the random variable X.
    • shape

      private final double shape
      The shape parameter of this distribution.
    • pdf

      private final DoubleUnaryOperator pdf
      Implementation of PDF(x). Assumes that x >= scale.
    • logpdf

      private final DoubleUnaryOperator logpdf
      Implementation of log PDF(x). Assumes that x >= scale.
  • Constructor Details

    • ParetoDistribution

      private ParetoDistribution(double scale, double shape)
      Parameters:
      scale - Scale parameter (minimum possible value of X).
      shape - Shape parameter (Pareto index).
  • 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
      Overrides:
      inverseCumulativeProbability in class AbstractContinuousDistribution
      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:

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

      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
      Overrides:
      inverseSurvivalProbability in class AbstractContinuousDistribution
      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
      Overrides:
      createSampler in class AbstractContinuousDistribution
      Parameters:
      rng - Generator of uniformly distributed numbers.
      Returns:
      a sampler that produces random numbers according this distribution.