SSJ
V. 2.6.

## umontreal.iro.lecuyer.probdist Class Pearson6Dist

```java.lang.Object
umontreal.iro.lecuyer.probdist.ContinuousDistribution
umontreal.iro.lecuyer.probdist.Pearson6Dist
```
All Implemented Interfaces:
Distribution

`public class Pearson6Distextends 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) = FB(x/(x + β))        for x > 0,

and F(x) = 0 otherwise, where FB(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
```Pearson6Dist(double alpha1, double alpha2, double beta)```
Constructs a Pearson6Dist object with parameters α1 = alpha1, α2 = alpha2 and β = beta.

Method Summary
` 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.
` 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.
` 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)]

### 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 String toString()`
Overrides:
`toString` in class `Object`

SSJ
V. 2.6.

To submit a bug or ask questions, send an e-mail to Pierre L'Ecuyer.