Class ParetoDistribution
- java.lang.Object
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- org.apache.commons.statistics.distribution.ParetoDistribution
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- 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.
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Nested Class Summary
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Nested classes/interfaces inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution
ContinuousDistribution.Sampler
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Method Summary
All Methods Static Methods Instance Methods Concrete Methods Modifier and Type Method Description ContinuousDistribution.SamplercreateSampler(org.apache.commons.rng.UniformRandomProvider rng)Creates a sampler.doublecumulativeProbability(double x)For a random variableXwhose values are distributed according to this distribution, this method returnsP(X <= x).doubledensity(double x)Returns the probability density function (PDF) of this distribution evaluated at the specified pointx.doublegetMean()Gets the mean of this distribution.doublegetScale()Gets the scale parameter of this distribution.doublegetShape()Gets the shape parameter of this distribution.doublegetSupportLowerBound()Gets the lower bound of the support.doublegetSupportUpperBound()Gets the upper bound of the support.doublegetVariance()Gets the variance of this distribution.doubleinverseCumulativeProbability(double p)Computes the quantile function of this distribution.doubleinverseSurvivalProbability(double p)Computes the inverse survival probability function of this distribution.doublelogDensity(double x)Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified pointx.static ParetoDistributionof(double scale, double shape)Creates a Pareto distribution.doubleprobability(double x0, double x1)For a random variableXwhose values are distributed according to this distribution, this method returnsP(x0 < X <= x1).doublesurvivalProbability(double x)For a random variableXwhose values are distributed according to this distribution, this method returnsP(X > x).
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Method Detail
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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- ifscale <= 0,scaleis infinite, orshape <= 0.
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getScale
public double getScale()
Gets the scale parameter of this distribution. This is the minimum possible value of X.- Returns:
- the scale parameter.
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getShape
public double getShape()
Gets the shape parameter of this distribution. This is the Pareto index.- Returns:
- the shape parameter.
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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 the CDF. 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.
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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.
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cumulativeProbability
public double cumulativeProbability(double x)
For a random variableXwhose 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.
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survivalProbability
public double survivalProbability(double x)
For a random variableXwhose 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.
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inverseCumulativeProbability
public double inverseCumulativeProbability(double p)
Computes the quantile function of this distribution. For a random variableXdistributed 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:
inverseCumulativeProbabilityin interfaceContinuousDistribution- Parameters:
p- Cumulative probability.- Returns:
- the smallest
p-quantile of this distribution (largest 0-quantile forp = 0).
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inverseSurvivalProbability
public double inverseSurvivalProbability(double p)
Computes the inverse survival probability function of this distribution. For a random variableXdistributed according to this distribution, the returned value is:\[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \gt x) \le p\} & \text{for } 0 \le p \lt 1 \\ \inf \{ x \in \mathbb R : P(X \gt x) \lt 1 \} & \text{for } p = 1 \end{cases} \]
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:
inverseSurvivalProbabilityin interfaceContinuousDistribution- Parameters:
p- Survival probability.- Returns:
- the smallest
(1-p)-quantile of this distribution (largest 0-quantile forp = 1).
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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.
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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.
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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.
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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.
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createSampler
public ContinuousDistribution.Sampler createSampler(org.apache.commons.rng.UniformRandomProvider rng)
Creates a sampler.- Specified by:
createSamplerin interfaceContinuousDistribution- Parameters:
rng- Generator of uniformly distributed numbers.- Returns:
- a sampler that produces random numbers according this distribution.
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probability
public double probability(double x0, double x1)For a random variableXwhose values are distributed according to this distribution, this method returnsP(x0 < X <= x1). The default implementation uses the identityP(x0 < X <= x1) = P(X <= x1) - P(X <= x0)- Specified by:
probabilityin interfaceContinuousDistribution- Parameters:
x0- Lower bound (exclusive).x1- Upper bound (inclusive).- Returns:
- the probability that a random variable with this distribution
takes a value between
x0andx1, excluding the lower and including the upper endpoint.
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