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,
• \u03b1 is the shape parameter: this is the Pareto index

See Also:

Pareto distribution (Wikipedia), Pareto distribution (MathWorld), Serialized Form

Field Summary

Fields inherited from class org.hipparchus.distribution.continuous.AbstractRealDistribution

DEFAULT_SOLVER_ABSOLUTE_ACCURACY

Constructor Summary

Constructors

Constructor	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)	Creates a Pareto distribution.

Method Summary

Modifier and Type	Method	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	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.
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.
double	logDensity(double x)	Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified point x.

Methods inherited from class org.hipparchus.distribution.continuous.AbstractRealDistribution

getSolverAbsoluteAccuracy, inverseCumulativeProbability, probability

Methods inherited from class java.lang.Object

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

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 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 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 \u03b1 of this distribution, the PDF is given by

• 0 if x < k,
• \u03b1 * k^\u03b1 / x^(\u03b1 + 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 \u03b1 of this distribution, the CDF is given by

• 0 if x < k,
• 1 - (k / x)^\u03b1 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 \u03b1, the mean is given by

• \u221e if \u03b1 <= 1,
• \u03b1 * k / (\u03b1 - 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 \u03b1, the variance is given by

• \u221e if 1 < \u03b1 <= 2,
• k^2 * \u03b1 / ((\u03b1 - 1)^2 * (\u03b1 - 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