Class ParetoDistribution
- All Implemented Interfaces:
ContinuousDistribution
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:
-
Nested Class Summary
Nested ClassesModifier and TypeClassDescriptionprivate static class
Create a RNG that inverts the output from nextDouble() as (1 - nextDouble()).Nested classes/interfaces inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution
ContinuousDistribution.Sampler
-
Field Summary
FieldsModifier and TypeFieldDescriptionprivate final DoubleUnaryOperator
Implementation of log PDF(x).private static final double
The minimum value for the shape parameter when computing when computing the variance.private final DoubleUnaryOperator
Implementation of PDF(x).private final double
The scale parameter of this distribution.private final double
The shape parameter of this distribution. -
Constructor Summary
Constructors -
Method Summary
Modifier and TypeMethodDescriptioncreateSampler
(org.apache.commons.rng.UniformRandomProvider rng) Creates a sampler.double
cumulativeProbability
(double x) For a random variableX
whose values are distributed according to this distribution, this method returnsP(X <= x)
.double
density
(double x) Returns the probability density function (PDF) of this distribution evaluated at the specified pointx
.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
Gets the lower bound of the support.double
Gets the upper bound of the support.double
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 pointx
.static ParetoDistribution
of
(double scale, double shape) Creates a Pareto distribution.double
survivalProbability
(double x) For a random variableX
whose values are distributed according to this distribution, this method returnsP(X > x)
.Methods inherited from class org.apache.commons.statistics.distribution.AbstractContinuousDistribution
getMedian, isSupportConnected, probability
-
Field Details
-
MIN_SHAPE_FOR_VARIANCE
private static final double MIN_SHAPE_FOR_VARIANCEThe minimum value for the shape parameter when computing when computing the variance.- See Also:
-
scale
private final double scaleThe scale parameter of this distribution. Also known ask
; the minimum possible value for the random variableX
. -
shape
private final double shapeThe shape parameter of this distribution. -
pdf
Implementation of PDF(x). Assumes thatx >= scale
. -
logpdf
Implementation of log PDF(x). Assumes thatx >= 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
Creates a Pareto distribution.- Parameters:
scale
- Scale parameter (minimum possible value of X).shape
- Shape parameter (Pareto index).- Returns:
- the distribution
- Throws:
IllegalArgumentException
- ifscale <= 0
,scale
is infinite, orshape <= 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 pointx
. In general, the PDF is the derivative of theCDF
. If the derivative does not exist atx
, 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 pointx
.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 variableX
whose values are distributed according to this distribution, this method returnsP(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 variableX
whose values are distributed according to this distribution, this method returnsP(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 variableX
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()
forp = 0
,ContinuousDistribution.getSupportUpperBound()
forp = 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 interfaceContinuousDistribution
- Overrides:
inverseCumulativeProbability
in classAbstractContinuousDistribution
- Parameters:
p
- Cumulative probability.- Returns:
- the smallest
p
-quantile of this distribution (largest 0-quantile forp = 0
).
-
inverseSurvivalProbability
public double inverseSurvivalProbability(double p) Computes the inverse survival probability function of this distribution. For a random variableX
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()
forp = 1
,ContinuousDistribution.getSupportUpperBound()
forp = 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 interfaceContinuousDistribution
- Overrides:
inverseSurvivalProbability
in classAbstractContinuousDistribution
- Parameters:
p
- Survival probability.- Returns:
- the smallest
(1-p)
-quantile of this distribution (largest 0-quantile forp = 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 asinverseCumulativeProbability(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 asinverseCumulativeProbability(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 interfaceContinuousDistribution
- Overrides:
createSampler
in classAbstractContinuousDistribution
- Parameters:
rng
- Generator of uniformly distributed numbers.- Returns:
- a sampler that produces random numbers according this distribution.
-