Pearson6Dist (Java Libraries for Stochastic Simulation)

java.lang.Object
- umontreal.iro.lecuyer.probdist.ContinuousDistribution
- - umontreal.iro.lecuyer.probdist.Pearson6Dist

All Implemented Interfaces:

Distribution
```
public class Pearson6Dist
extends ContinuousDistribution
```
Extends the class ContinuousDistribution for the Pearson type VI distribution with shape parameters α₁ > 0 and α₂ > 0, and scale parameter β > 0. The density function is given by

f (x) = (x/β)^α₁-1/(βB(α₁, α₂)[1 + x/β]^α₁+α₂) for x > 0,
and f (x) = 0 otherwise, where B is the beta function. The distribution function is given by

F(x) = F_B(x/(x + β)) for x > 0,
and F(x) = 0 otherwise, where F_B(x) is the distribution function of a beta distribution with shape parameters α₁ and α₂.

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

Constructor Summary

Constructors
Constructor and Description
`Pearson6Dist(double alpha1, double alpha2, double beta)` Constructs a `Pearson6Dist` object with parameters α₁ = `alpha1`, α₂ = `alpha2` 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 alpha1, double alpha2, double beta, double x)` Computes the complementary distribution function of a Pearson VI distribution with shape parameters α₁ and α₂, and scale parameter β.
`double`	`cdf(double x)` Returns the distribution function F(x).
`static double`	`cdf(double alpha1, double alpha2, double beta, double x)` Computes the distribution function of a Pearson VI distribution with shape parameters α₁ and α₂, and scale parameter β.
`double`	`density(double x)` Returns f (x), the density evaluated at x.
`static double`	`density(double alpha1, double alpha2, double beta, double x)` Computes the density function of a Pearson VI distribution with shape parameters α₁ and α₂, and scale parameter β.
`double`	`getAlpha1()` Returns the α₁ parameter of this object.
`double`	`getAlpha2()` Returns the α₂ parameter of this object.
`double`	`getBeta()` Returns the β parameter of this object.
`static Pearson6Dist`	`getInstanceFromMLE(double[] x, int n)` Creates a new instance of a Pearson VI distribution with parameters α₁, α₂ 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 alpha1, double alpha2, double beta)` Computes and returns the mean E[X] = (βα₁)/(α₂ - 1) of a Pearson VI distribution with shape parameters α₁ and α₂, and scale parameter β.
`static double[]`	`getMLE(double[] x, int n)` Estimates the parameters (α₁, α₂, β) of the Pearson VI 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 alpha1, double alpha2, double beta)` Computes and returns the standard deviation of a Pearson VI distribution with shape parameters α₁ and α₂, and scale parameter β.
`double`	`getVariance()` Returns the variance.
`static double`	`getVariance(double alpha1, double alpha2, double beta)` Computes and returns the variance Var[X] = [β²α₁(α₁ + α₂ -1)]/[(α₂ -1)²(α₂ - 2)] of a Pearson VI distribution with shape parameters α₁ and α₂, and scale parameter β.
`double`	`inverseF(double u)` Returns the inverse distribution function x = F^-1(u).
`static double`	`inverseF(double alpha1, double alpha2, double beta, double u)` Computes the inverse distribution function of a Pearson VI distribution with shape parameters α₁ and α₂, and scale parameter β.
`void`	`setParam(double alpha1, double alpha2, double beta)` Sets the parameters α₁, α₂ and β of 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
  - Pearson6Dist
```
public Pearson6Dist(double alpha1,
            double alpha2,
            double beta)
```
    Constructs a Pearson6Dist object with parameters α₁ = alpha1, α₂ = alpha2 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 alpha1,
             double alpha2,
             double beta,
             double x)
```
    Computes the density function of a Pearson VI distribution with shape parameters α₁ and α₂, and scale parameter β.
  - cdf
```
public static double cdf(double alpha1,
         double alpha2,
         double beta,
         double x)
```
    Computes the distribution function of a Pearson VI distribution with shape parameters α₁ and α₂, and scale parameter β.
  - barF
```
public static double barF(double alpha1,
          double alpha2,
          double beta,
          double x)
```
    Computes the complementary distribution function of a Pearson VI distribution with shape parameters α₁ and α₂, and scale parameter β.
  - inverseF
```
public static double inverseF(double alpha1,
              double alpha2,
              double beta,
              double u)
```
    Computes the inverse distribution function of a Pearson VI distribution with shape parameters α₁ and α₂, and scale parameter β.
  - getMLE
```
public static double[] getMLE(double[] x,
              int n)
```
    Estimates the parameters (α₁, α₂, β) of the Pearson VI distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1. The estimates are returned in a three-element array, in regular order: [ α₁, α₂, β].
    
    Parameters:
    x - the list of observations to use to evaluate parameters
    n - the number of observations to use to evaluate parameters
    
    Returns:
    returns the parameters [ hat(α_1), hat(α_2), hat(β)]
  - getMaximumLikelihoodEstimate
```
@Deprecated
public static double[] getMaximumLikelihoodEstimate(double[] x,
                                               int n)
```
    Deprecated.
    
    Same as getMLE.
  - getInstanceFromMLE
```
public static Pearson6Dist getInstanceFromMLE(double[] x,
                              int n)
```
    Creates a new instance of a Pearson VI distribution with parameters α₁, α₂ 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 alpha1,
             double alpha2,
             double beta)
```
    Computes and returns the mean E[X] = (βα₁)/(α₂ - 1) of a Pearson VI distribution with shape parameters α₁ and α₂, and scale parameter β.
  - getVariance
```
public static double getVariance(double alpha1,
                 double alpha2,
                 double beta)
```
    Computes and returns the variance Var[X] = [β²α₁(α₁ + α₂ -1)]/[(α₂ -1)²(α₂ - 2)] of a Pearson VI distribution with shape parameters α₁ and α₂, and scale parameter β.
  - getStandardDeviation
```
public static double getStandardDeviation(double alpha1,
                          double alpha2,
                          double beta)
```
    Computes and returns the standard deviation of a Pearson VI distribution with shape parameters α₁ and α₂, and scale parameter β.
  - getAlpha1
```
public double getAlpha1()
```
    Returns the α₁ parameter of this object.
  - getAlpha2
```
public double getAlpha2()
```
    Returns the α₂ parameter of this object.
  - getBeta
```
public double getBeta()
```
    Returns the β parameter of this object.
  - setParam
```
public void setParam(double alpha1,
            double alpha2,
            double beta)
```
    Sets the parameters α₁, α₂ and β of this object.
  - getParams
```
public double[] getParams()
```
    Return a table containing the parameters of the current distribution. This table is put in regular order: [α₁, α₂, β].
  - toString
```
public java.lang.String toString()
```
    Overrides:
    
    toString in class java.lang.Object

Class Pearson6Dist

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

Pearson6Dist

Method Detail

density

cdf

barF

inverseF

getMean

getVariance

getStandardDeviation

density

cdf

barF

inverseF

getMLE

getMaximumLikelihoodEstimate

getInstanceFromMLE

getMean

getVariance

getStandardDeviation

getAlpha1

getAlpha2

getBeta

setParam

getParams

toString