SSJ
V. 1.2.5.

umontreal.iro.lecuyer.probdist
Class Pearson6Dist

java.lang.Object
  extended by umontreal.iro.lecuyer.probdist.ContinuousDistribution
      extended by 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 bar(F)(x) = 1 - F(x).
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)
          Computes and 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 of X 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 in table x[i], i = 0, 1,…, n - 1.
static double[] getMaximumLikelihoodEstimate(double[] x, int n)
          Estimates and returns the parameters [ hat(α_1), hat(α_2), hat(β)] of the Pearson VI distribution using the maximum likelihood method based on the n observations in table x[i], i = 0, 1,…, n - 1.
 double getMean()
          Returns the mean of the distribution function.
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.
 double getStandardDeviation()
          Returns the standard deviation of the distribution function.
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 of the distribution function.
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 F-1(u), defined in.
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.
 
Methods inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
inverseBisection, inverseBrent
 
Methods inherited from class java.lang.Object
equals, getClass, hashCode, notify, notifyAll, toString, 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 of X 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
Computes and 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 interface: Distribution
Returns 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 interface: Distribution
Computes and returns the inverse distribution function F-1(u), defined in.

Specified by:
inverseF in interface Distribution
Overrides:
inverseF in class ContinuousDistribution
Parameters:
u - value in the interval (0, 1) for which the inverse distribution function is evaluated
Returns:
the inverse distribution function evaluated at u

getMean

public double getMean()
Description copied from interface: Distribution
Returns the mean of the distribution function.


getVariance

public double getVariance()
Description copied from interface: Distribution
Returns the variance of the distribution function.


getStandardDeviation

public double getStandardDeviation()
Description copied from interface: Distribution
Returns the standard deviation of the distribution function.


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.


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 in table 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

getMaximumLikelihoodEstimate

public static double[] getMaximumLikelihoodEstimate(double[] x,
                                                    int n)
Estimates and returns the parameters [ hat(α_1), hat(α_2), hat(β)] of the Pearson VI distribution using the maximum likelihood method based on the n observations in table 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
Returns:
returns the parameters [ hat(α_1), hat(α_2), hat(β)]

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.


SSJ
V. 1.2.5.

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