ParetoDist (Java Libraries for Stochastic Simulation)

java.lang.Object
- umontreal.iro.lecuyer.probdist.ContinuousDistribution
- - umontreal.iro.lecuyer.probdist.ParetoDist

All Implemented Interfaces:

Distribution
```
public class ParetoDist
extends ContinuousDistribution
```
Extends the class ContinuousDistribution for a distribution from the Pareto family, with shape parameter α > 0 and location parameter β > 0. The density for this type of Pareto distribution is

f (x) = αβ^α/x^α+1        for x >= β,
and 0 otherwise. The distribution function is

F(x) = 1 - (β/x)^α        for x >= β,
and the inverse distribution function is

F^-1(u) = β(1 - u)^-1/α        for 0 <= u < 1.

Field Summary
- Fields inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
  decPrec

Constructor Summary

Constructors
Constructor and Description
`ParetoDist(double alpha)` Constructs a `ParetoDist` object with parameters α = `alpha` and β = 1.
`ParetoDist(double alpha, double beta)` Constructs a `ParetoDist` object with parameters α = `alpha` and β = `beta`.

Method Summary

Methods
Modifier and Type	Method and Description
`double`	`barF(double x)` Returns the complementary distribution function.
`static double`	`barF(double alpha, double beta, double x)` Computes the complementary distribution function.
`double`	`cdf(double x)` Returns the distribution function F(x).
`static double`	`cdf(double alpha, double beta, double x)` Computes the distribution function.
`double`	`density(double x)` Returns f (x), the density evaluated at x.
`static double`	`density(double alpha, double beta, double x)` Computes the density function.
`double`	`getAlpha()` Returns the parameter α.
`double`	`getBeta()` Returns the parameter β.
`static ParetoDist`	`getInstanceFromMLE(double[] x, int n)` Creates a new instance of a Pareto distribution with parameters α and β estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.
`static double[]`	`getMaximumLikelihoodEstimate(double[] x, int n)` Deprecated.
`double`	`getMean()` Returns the mean.
`static double`	`getMean(double alpha, double beta)` Computes and returns the mean E[X] = αβ/(α - 1) of the Pareto distribution with parameters α and β.
`static double[]`	`getMLE(double[] x, int n)` Estimates the parameters (α, β) of the Pareto distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1.
`double[]`	`getParams()` Return a table containing the parameters of the current distribution.
`double`	`getStandardDeviation()` Returns the standard deviation.
`static double`	`getStandardDeviation(double alpha, double beta)` Computes and returns the standard deviation of the Pareto distribution with parameters α and β.
`double`	`getVariance()` Returns the variance.
`static double`	`getVariance(double alpha, double beta)` Computes and returns the variance of the Pareto distribution with parameters α and β.
`double`	`inverseF(double u)` Returns the inverse distribution function x = F^-1(u).
`static double`	`inverseF(double alpha, double beta, double u)` Computes the inverse of the distribution function.
`void`	`setParams(double alpha, double beta)` Sets the parameter α and β for this object.
`java.lang.String`	`toString()`

Methods inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
getXinf, getXsup, inverseBisection, inverseBrent, setXinf, setXsup

Methods inherited from class java.lang.Object
equals, getClass, hashCode, notify, notifyAll, wait, wait, wait

- Constructor Detail
  - ParetoDist
```
public ParetoDist(double alpha)
```
    Constructs a ParetoDist object with parameters α = alpha and β = 1.
  - ParetoDist
```
public ParetoDist(double alpha,
          double beta)
```
    Constructs a ParetoDist object with parameters α = alpha and β = beta.
- Method Detail
  - density
```
public double density(double x)
```
    Description copied from class: ContinuousDistribution
    
    Returns f (x), the density evaluated at x.
    
    Specified by:
    
    density in class ContinuousDistribution
    
    Parameters:
    x - value at which the density is evaluated
    
    Returns:
    density function evaluated at x
  - cdf
```
public double cdf(double x)
```
    Description copied from interface: Distribution
    
    Returns the distribution function F(x).
    
    Parameters:
    x - value at which the distribution function is evaluated
    
    Returns:
    distribution function evaluated at x
  - barF
```
public double barF(double x)
```
    Description copied from class: ContinuousDistribution
    
    Returns the complementary distribution function. The default implementation computes bar(F)(x) = 1 - F(x).
    
    Specified by:
    
    barF in interface Distribution
    
    Overrides:
    
    barF in class ContinuousDistribution
    
    Parameters:
    x - value at which the complementary distribution function is evaluated
    
    Returns:
    complementary distribution function evaluated at x
  - inverseF
```
public double inverseF(double u)
```
    Description copied from class: ContinuousDistribution
    
    Returns the inverse distribution function x = F^-1(u). Restrictions: u∈[0, 1].
    
    Specified by:
    
    inverseF in interface Distribution
    
    Overrides:
    
    inverseF in class ContinuousDistribution
    
    Parameters:
    u - value at which the inverse distribution function is evaluated
    
    Returns:
    the inverse distribution function evaluated at u
  - getMean
```
public double getMean()
```
    Description copied from class: ContinuousDistribution
    
    Returns the mean.
    
    Specified by:
    
    getMean in interface Distribution
    
    Overrides:
    
    getMean in class ContinuousDistribution
    
    Returns:
    the mean
  - getVariance
```
public double getVariance()
```
    Description copied from class: ContinuousDistribution
    
    Returns the variance.
    
    Specified by:
    
    getVariance in interface Distribution
    
    Overrides:
    
    getVariance in class ContinuousDistribution
    
    Returns:
    the variance
  - getStandardDeviation
```
public double getStandardDeviation()
```
    Description copied from class: ContinuousDistribution
    
    Returns the standard deviation.
    
    Specified by:
    
    getStandardDeviation in interface Distribution
    
    Overrides:
    
    getStandardDeviation in class ContinuousDistribution
    
    Returns:
    the standard deviation
  - density
```
public static double density(double alpha,
             double beta,
             double x)
```
    Computes the density function.
  - cdf
```
public static double cdf(double alpha,
         double beta,
         double x)
```
    Computes the distribution function.
  - barF
```
public static double barF(double alpha,
          double beta,
          double x)
```
    Computes the complementary distribution function.
  - inverseF
```
public static double inverseF(double alpha,
              double beta,
              double u)
```
    Computes the inverse of the distribution function.
  - getMLE
```
public static double[] getMLE(double[] x,
              int n)
```
    Estimates the parameters (α, β) of the Pareto distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1. The estimates are returned in a two-element array, in regular order: [α, β].
    
    Parameters:
    x - the list of observations used to evaluate parameters
    n - the number of observations used to evaluate parameters
    
    Returns:
    returns the parameters [ hat(α), hat(β)]
  - getMaximumLikelihoodEstimate
```
@Deprecated
public static double[] getMaximumLikelihoodEstimate(double[] x,
                                               int n)
```
    Deprecated.
    
    Same as getMLE.
  - getInstanceFromMLE
```
public static ParetoDist getInstanceFromMLE(double[] x,
                            int n)
```
    Creates a new instance of a Pareto distribution with parameters α and β estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.
    
    Parameters:
    x - the list of observations to use to evaluate parameters
    n - the number of observations to use to evaluate parameters
  - getMean
```
public static double getMean(double alpha,
             double beta)
```
    Computes and returns the mean E[X] = αβ/(α - 1) of the Pareto distribution with parameters α and β.
    
    Returns:
    the mean of the Pareto distribution
  - getVariance
```
public static double getVariance(double alpha,
                 double beta)
```
    Computes and returns the variance of the Pareto distribution with parameters α and β.
    
    Returns:
    the variance of the Pareto distribution Var[X] = αβ²/[(α -2)(α - 1)]
  - getStandardDeviation
```
public static double getStandardDeviation(double alpha,
                          double beta)
```
    Computes and returns the standard deviation of the Pareto distribution with parameters α and β.
    
    Returns:
    the standard deviation of the Pareto distribution
  - getAlpha
```
public double getAlpha()
```
    Returns the parameter α.
  - getBeta
```
public double getBeta()
```
    Returns the parameter β.
  - setParams
```
public void setParams(double alpha,
             double beta)
```
    Sets the parameter α and β for this object.
  - getParams
```
public double[] getParams()
```
    Return a table containing the parameters of the current distribution. This table is put in regular order: [α, β].
  - toString
```
public java.lang.String toString()
```
    Overrides:
    
    toString in class java.lang.Object

Class ParetoDist

Field Summary

Fields inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution

Constructor Summary

Method Summary

Methods inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution

Methods inherited from class java.lang.Object

Constructor Detail

ParetoDist

ParetoDist

Method Detail

density

cdf

barF

inverseF

getMean

getVariance

getStandardDeviation

density

cdf

barF

inverseF

getMLE

getMaximumLikelihoodEstimate

getInstanceFromMLE

getMean

getVariance

getStandardDeviation

getAlpha

getBeta

setParams

getParams

toString