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## 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 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) = 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.
`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)` Computes and returns the inverse distribution function x = F-1(u), by calling the method `inverseBrent`.
`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
`inverseBisection, inverseBrent`

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), which is not accurate when F(x) -> 1.

  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`
  Computes and returns the inverse distribution function x = F-1(u), by calling the method `inverseBrent`. 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 String toString()

  Overrides:
    `toString` in class `Object`

Overview | Package | Class | Tree | Deprecated | Index | Help
| SSJ V. 2.1.1.
PREV CLASS NEXT CLASS | FRAMES NO FRAMES
SUMMARY: NESTED | FIELD | CONSTR | METHOD | DETAIL: FIELD | CONSTR | METHOD

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