ParetoDistribution (Apache Commons Statistics Distribution 1.0 API)

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:

Pareto distribution (Wikipedia), Pareto distribution (MathWorld)

Nested Class Summary
- Nested classes/interfaces inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution
  ContinuousDistribution.Sampler

Method Summary

All Methods Static Methods Instance Methods Concrete Methods
Modifier and Type	Method and Description
`ContinuousDistribution.Sampler`	`createSampler(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 Detail
  - 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:
    - ContinuousDistribution.getSupportLowerBound() for p = 0,
    - ContinuousDistribution.getSupportUpperBound() for p = 1, or
    - the result of a search for a root between the lower and upper bound using cumulativeProbability(x) - p. The bounds may be bracketed for efficiency.
    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:
    \[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \ge x) \le p\} & \text{for } 0 \le p \lt 1 \\ \inf \{ x \in \mathbb R : P(X \ge x) \lt 1 \} & \text{for } p = 1 \end{cases} \]
    By default, this is defined as inverseCumulativeProbability(1 - p), but the specific implementation may be more accurate.
    The default implementation returns:
    - ContinuousDistribution.getSupportLowerBound() for p = 1,
    - ContinuousDistribution.getSupportUpperBound() for p = 0, or
    - the result of a search for a root between the lower and upper bound using survivalProbability(x) - p. The bounds may be bracketed for efficiency.
    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(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.

Class ParetoDistribution

Nested Class Summary

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

Method Summary

Methods inherited from class java.lang.Object

Method Detail

of

getScale

getShape

density

logDensity

cumulativeProbability

survivalProbability

inverseCumulativeProbability

inverseSurvivalProbability

getMean

getVariance

getSupportLowerBound

getSupportUpperBound

createSampler

probability