ParetoDistribution (Apache Commons Math 3.6.1 API)

org.apache.commons.math3.distribution

## Class ParetoDistribution

- java.lang.Object
- - org.apache.commons.math3.distribution.AbstractRealDistribution
  - - org.apache.commons.math3.distribution.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
  Since:
    3.3
  See Also:
    Pareto distribution (Wikipedia), Pareto distribution (MathWorld), Serialized Form

- - ### Field Summary

    Fields
    Modifier and Type | Field and Description
    --- | ---
    `static double` | `DEFAULT_INVERSE_ABSOLUTE_ACCURACY` Default inverse cumulative probability accuracy.

    - ### Fields inherited from class org.apache.commons.math3.distribution.AbstractRealDistribution

      `random, randomData, SOLVER_DEFAULT_ABSOLUTE_ACCURACY`

  - ### Constructor Summary

    Constructors
    Constructor and Description
    `ParetoDistribution()` Create a Pareto distribution with a scale of `1` and a shape of `1`.
    `ParetoDistribution(double scale, double shape)` Create a Pareto distribution using the specified scale and shape.
    `ParetoDistribution(double scale, double shape, double inverseCumAccuracy)` Create a Pareto distribution using the specified scale, shape and inverse cumulative distribution accuracy.
    `ParetoDistribution(RandomGenerator rng, double scale, double shape)` Creates a Pareto distribution.
    `ParetoDistribution(RandomGenerator rng, double scale, double shape, double inverseCumAccuracy)` Creates a Pareto distribution.

  - ### Method Summary

    Methods
    Modifier and Type | Method and Description
    --- | ---
    `double` | `cumulativeProbability(double x)` For a random variable `X` whose values are distributed according to this distribution, this method returns `P(X <= x)`.
    `double` | `cumulativeProbability(double x0, double x1)` Deprecated. See `RealDistribution.cumulativeProbability(double,double)`
    `double` | `density(double x)` Returns the probability density function (PDF) of this distribution evaluated at the specified point `x`.
    `double` | `getNumericalMean()` Use this method to get the numerical value of the mean of this distribution.
    `double` | `getNumericalVariance()` Use this method to get the numerical value of the variance of this distribution.
    `double` | `getScale()` Returns the scale parameter of this distribution.
    `double` | `getShape()` Returns the shape parameter of this distribution.
    `protected double` | `getSolverAbsoluteAccuracy()` Returns the solver absolute accuracy for inverse cumulative computation.
    `double` | `getSupportLowerBound()` Access the lower bound of the support.
    `double` | `getSupportUpperBound()` Access the upper bound of the support.
    `boolean` | `isSupportConnected()` Use this method to get information about whether the support is connected, i.e.
    `boolean` | `isSupportLowerBoundInclusive()` Whether or not the lower bound of support is in the domain of the density function.
    `boolean` | `isSupportUpperBoundInclusive()` Whether or not the upper bound of support is in the domain of the density function.
    `double` | `logDensity(double x)` Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified point `x`.
    `double` | `sample()` Generate a random value sampled from this distribution.

    - ### Methods inherited from class org.apache.commons.math3.distribution.AbstractRealDistribution

      `inverseCumulativeProbability, probability, probability, reseedRandomGenerator, sample`

    - ### Methods inherited from class java.lang.Object

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

- - ### Field Detail

    - #### DEFAULT_INVERSE_ABSOLUTE_ACCURACY

          public static final double DEFAULT_INVERSE_ABSOLUTE_ACCURACY

      Default inverse cumulative probability accuracy.
      See Also:
        Constant Field Values

  - ### Constructor Detail

    - #### 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 NotStrictlyPositiveException

      Create a Pareto distribution using the specified scale and shape.

      Note: this constructor will implicitly create an instance of `Well19937c` as random generator to be used for sampling only (see `sample()` and `AbstractRealDistribution.sample(int)`). In case no sampling is needed for the created distribution, it is advised to pass `null` as random generator via the appropriate constructors to avoid the additional initialisation overhead.

      Parameters:
        `scale` - the scale parameter of this distribution
        `shape` - the shape parameter of this distribution
      Throws:
        `NotStrictlyPositiveException` - if `scale <= 0` or `shape <= 0`.

    - #### ParetoDistribution

          public ParetoDistribution(double scale,
                            double shape,
                            double inverseCumAccuracy)
                             throws NotStrictlyPositiveException

      Create a Pareto distribution using the specified scale, shape and inverse cumulative distribution accuracy.

      Note: this constructor will implicitly create an instance of `Well19937c` as random generator to be used for sampling only (see `sample()` and `AbstractRealDistribution.sample(int)`). In case no sampling is needed for the created distribution, it is advised to pass `null` as random generator via the appropriate constructors to avoid the additional initialisation overhead.

      Parameters:
        `scale` - the scale parameter of this distribution
        `shape` - the shape parameter of this distribution
        `inverseCumAccuracy` - Inverse cumulative probability accuracy.
      Throws:
        `NotStrictlyPositiveException` - if `scale <= 0` or `shape <= 0`.

    - #### ParetoDistribution

          public ParetoDistribution(RandomGenerator rng,
                            double scale,
                            double shape)
                             throws NotStrictlyPositiveException

      Creates a Pareto distribution.
      Parameters:
        `rng` - Random number generator.
        `scale` - Scale parameter of this distribution.
        `shape` - Shape parameter of this distribution.
      Throws:
        `NotStrictlyPositiveException` - if `scale <= 0` or `shape <= 0`.

    - #### ParetoDistribution

          public ParetoDistribution(RandomGenerator rng,
                            double scale,
                            double shape,
                            double inverseCumAccuracy)
                             throws NotStrictlyPositiveException

      Creates a Pareto distribution.
      Parameters:
        `rng` - Random number generator.
        `scale` - Scale parameter of this distribution.
        `shape` - Shape parameter of this distribution.
        `inverseCumAccuracy` - Inverse cumulative probability accuracy.
      Throws:
        `NotStrictlyPositiveException` - 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` - 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.
      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`

    - #### cumulativeProbability

          @Deprecated
          public double cumulativeProbability(double x0,
                                                double x1)
                                       throws NumberIsTooLargeException

      Deprecated. See `RealDistribution.cumulativeProbability(double,double)`
      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:
        `cumulativeProbability` in interface `RealDistribution`
      Overrides:
        `cumulativeProbability` in class `AbstractRealDistribution`
      Parameters:
        `x0` - the exclusive lower bound
        `x1` - the inclusive upper bound
      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
      Throws:
        `NumberIsTooLargeException` - if `x0 > x1`

    - #### getSolverAbsoluteAccuracy

          protected double getSolverAbsoluteAccuracy()

      Returns the solver absolute accuracy for inverse cumulative computation. You can override this method in order to use a Brent solver with an absolute accuracy different from the default.
      Overrides:
        `getSolverAbsoluteAccuracy` in class `AbstractRealDistribution`
      Returns:
        the maximum absolute error in inverse cumulative probability estimates

    - #### 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`)

    - #### isSupportLowerBoundInclusive

          public boolean isSupportLowerBoundInclusive()

      Whether or not the lower bound of support is in the domain of the density function. Returns true iff `getSupporLowerBound()` is finite and `density(getSupportLowerBound())` returns a non-NaN, non-infinite value.
      Returns:
        true if the lower bound of support is finite and the density function returns a non-NaN, non-infinite value there

    - #### isSupportUpperBoundInclusive

          public boolean isSupportUpperBoundInclusive()

      Whether or not the upper bound of support is in the domain of the density function. Returns true iff `getSupportUpperBound()` is finite and `density(getSupportUpperBound())` returns a non-NaN, non-infinite value.
      Returns:
        true if the upper bound of support is finite and the density function returns a non-NaN, non-infinite value there

    - #### 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`

    - #### sample

          public double sample()

      Generate a random value sampled from this distribution. The default implementation uses the inversion method.
      Specified by:
        `sample` in interface `RealDistribution`
      Overrides:
        `sample` in class `AbstractRealDistribution`
      Returns:
        a random value.

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