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) = 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

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.
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)]

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