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 $x\in [k,\mathrm{\infty })$.

$k$ 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

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 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:

$$f(x;k,\alpha )=\{\begin{array}{ll}0& \text{for}x<k\\ \frac{\alpha k^{\alpha }}{x^{\alpha +1}}& \text{for}x\ge k\end{array}$$

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:

$$F(x;k,\alpha )=\{\begin{array}{ll}0& \text{for}x\le k\\ 1-{\left(\frac{k}{x}\right)}^{\alpha }& \text{for}x>k\end{array}$$

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:

$$S(x;k,\alpha )=\{\begin{array}{ll}1& \text{for}x\le k\\ {\left(\frac{k}{x}\right)}^{\alpha }& \text{for}x>k\end{array}$$

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:

$$x=\{\begin{array}{ll}inf\{x\in \mathbb{R}:P(X\le x)\ge p\}& \text{for}0<p\le 1\\ inf\{x\in \mathbb{R}:P(X\le x)>0\}& \text{for}p=0\end{array}$$

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:

$$x=\{\begin{array}{ll}inf\{x\in \mathbb{R}:P(X>x)\le p\}& \text{for}0\le p<1\\ inf\{x\in \mathbb{R}:P(X>x)<1\}& \text{for}p=1\end{array}$$

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:

$$\mathbb{E}[X]=\{\begin{array}{ll}\mathrm{\infty }& \text{for}\alpha \le 1\\ \frac{k\alpha }{(\alpha -1)}& \text{for}\alpha >1\end{array}$$

Returns:

the mean.

getVariance

public double getVariance()

Gets the variance of this distribution.

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

$$\mathrm{var}[X]=\{\begin{array}{ll}\mathrm{\infty }& \text{for}\alpha \le 2\\ \frac{k^2\alpha }{(\alpha -1{)}^2(\alpha -2)}& \text{for}\alpha >2\end{array}$$

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. $inf\{x\in \mathbb{R}:P(X\le x)>0\}$.

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.