Extends the class ContinuousDistribution for the Pearson type VI distribution with shape parameters \(\alpha_1 > 0\) and \(\alpha_2 > 0\), and scale parameter \(\beta> 0\). More...

Inheritance diagram for umontreal.ssj.probdist.Pearson6Dist:

Public Member Functions
	Pearson6Dist (double alpha1, double alpha2, double beta)
	Constructs a Pearson6Dist object with parameters \(\alpha_1\) = alpha1, \(\alpha_2\) = alpha2 and \(\beta\) = beta.
double	density (double x)
	Returns \(f(x)\), the density evaluated at \(x\).
double	cdf (double x)
	Returns the distribution function \(F(x)\).
double	barF (double x)
	Returns the complementary distribution function.
double	inverseF (double u)
	Returns the inverse distribution function \(x = F^{-1}(u)\).
double	getMean ()
	Returns the mean.
double	getVariance ()
	Returns the variance.
double	getStandardDeviation ()
	Returns the standard deviation.
double	getAlpha1 ()
	Returns the \(\alpha_1\) parameter of this object.
double	getAlpha2 ()
	Returns the \(\alpha_2\) parameter of this object.
double	getBeta ()
	Returns the \(\beta\) parameter of this object.
void	setParam (double alpha1, double alpha2, double beta)
	Sets the parameters \(\alpha_1\), \(\alpha_2\) and.
double[]	getParams ()
	Return a table containing the parameters of the current distribution.
String	toString ()
	Returns a String containing information about the current distribution.
Public Member Functions inherited from umontreal.ssj.probdist.ContinuousDistribution
double	inverseBrent (double a, double b, double u, double tol)
	Computes the inverse distribution function \(x = F^{-1}(u)\), using the Brent-Dekker method.
double	inverseBisection (double u)
	Computes and returns the inverse distribution function \(x = F^{-1}(u)\), using bisection.
double	getXinf ()
	Returns \(x_a\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).
double	getXsup ()
	Returns \(x_b\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).
void	setXinf (double xa)
	Sets the value \(x_a=\) xa, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).
void	setXsup (double xb)
	Sets the value \(x_b=\) xb, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).

Static Public Member Functions
static double	density (double alpha1, double alpha2, double beta, double x)
	Computes the density function of a Pearson VI distribution with shape parameters \(\alpha_1\) and \(\alpha_2\), and scale parameter \(\beta\).
static double	cdf (double alpha1, double alpha2, double beta, double x)
	Computes the distribution function of a Pearson VI distribution with shape parameters \(\alpha_1\) and \(\alpha_2\), and scale parameter \(\beta\).
static double	barF (double alpha1, double alpha2, double beta, double x)
	Computes the complementary distribution function of a Pearson VI distribution with shape parameters \(\alpha_1\) and.
static double	inverseF (double alpha1, double alpha2, double beta, double u)
	Computes the inverse distribution function of a Pearson VI distribution with shape parameters \(\alpha_1\) and.
static double[]	getMLE (double[] x, int n)
	Estimates the parameters \((\alpha_1,\alpha_2,\beta)\) of the Pearson VI distribution using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\).
static Pearson6Dist	getInstanceFromMLE (double[] x, int n)
	Creates a new instance of a Pearson VI distribution with parameters.
static double	getMean (double alpha1, double alpha2, double beta)
	Computes and returns the mean \(E[X] = (\beta\alpha_1) / (\alpha_2 - 1)\) of a Pearson VI distribution with shape parameters \(\alpha_1\) and \(\alpha_2\), and scale parameter \(\beta\).
static double	getVariance (double alpha1, double alpha2, double beta)
	Computes and returns the variance \(\mbox{Var}[X] = [\beta^2 \alpha_1 (\alpha_1 + \alpha_2 - 1)] / [(\alpha_2 - 1)^2(\alpha_2 - 2)]\) of a Pearson VI distribution with shape parameters \(\alpha_1\) and \(\alpha_2\), and scale parameter \(\beta\).
static double	getStandardDeviation (double alpha1, double alpha2, double beta)
	Computes and returns the standard deviation of a Pearson VI distribution with shape parameters \(\alpha_1\) and.

Detailed Description

Extends the class ContinuousDistribution for the Pearson type VI distribution with shape parameters \(\alpha_1 > 0\) and \(\alpha_2 > 0\), and scale parameter \(\beta> 0\).

The density function is given by

\[ f(x) =\left\{\begin{array}{ll} \displaystyle\frac{\left(x/{\beta}\right)^{\alpha_1 - 1}}{\beta\mathcal{B}(\alpha_1, \alpha_2)(1 + x/{\beta})^{\alpha_1 + \alpha_2}} & \quad\mbox{for } x > 0, \\ 0 & \quad\mbox{otherwise,} \end{array} \right. \tag{fpearson6} \]

where \(\mathcal{B}\) is the beta function. The distribution function is given by

\[ F(x) = F_B\left(\frac{x}{x + \beta}\right) \qquad\mbox{for } x > 0, \tag{Fpearson6} \]

and \(F(x) = 0\) otherwise, where \(F_B(x)\) is the distribution function of a beta distribution with shape parameters \(\alpha_1\) and \(\alpha_2\).

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Definition at line 53 of file Pearson6Dist.java.

Constructor & Destructor Documentation

◆ Pearson6Dist()

umontreal.ssj.probdist.Pearson6Dist.Pearson6Dist	(	double	alpha1,
		double	alpha2,
		double	beta )

Constructs a Pearson6Dist object with parameters \(\alpha_1\) = alpha1, \(\alpha_2\) = alpha2 and \(\beta\) = beta.

Definition at line 99 of file Pearson6Dist.java.

Member Function Documentation

◆ barF() [1/2]

double umontreal.ssj.probdist.Pearson6Dist.barF	(	double	alpha1,
		double	alpha2,
		double	beta,
		double	x )

static

Computes the complementary distribution function of a Pearson VI distribution with shape parameters \(\alpha_1\) and.

\(\alpha_2\), and scale parameter \(\beta\).

Definition at line 177 of file Pearson6Dist.java.

◆ barF() [2/2]

double umontreal.ssj.probdist.Pearson6Dist.barF ( double x )

Returns the complementary distribution function.

The default implementation computes \(\bar{F}(x) = 1 - F(x)\).

Parameters

x	value at which the complementary distribution function is evaluated

Returns: complementary distribution function evaluated at x

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 114 of file Pearson6Dist.java.

◆ cdf() [1/2]

double umontreal.ssj.probdist.Pearson6Dist.cdf	(	double	alpha1,
		double	alpha2,
		double	beta,
		double	x )

static

Computes the distribution function of a Pearson VI distribution with shape parameters \(\alpha_1\) and \(\alpha_2\), and scale parameter \(\beta\).

Definition at line 158 of file Pearson6Dist.java.

◆ cdf() [2/2]

double umontreal.ssj.probdist.Pearson6Dist.cdf ( double x )

Returns the distribution function \(F(x)\).

Parameters

x	value at which the distribution function is evaluated

Returns: distribution function evaluated at x

Implements umontreal.ssj.probdist.Distribution.

Definition at line 110 of file Pearson6Dist.java.

◆ density() [1/2]

double umontreal.ssj.probdist.Pearson6Dist.density	(	double	alpha1,
		double	alpha2,
		double	beta,
		double	x )

static

Computes the density function of a Pearson VI distribution with shape parameters \(\alpha_1\) and \(\alpha_2\), and scale parameter \(\beta\).

Definition at line 139 of file Pearson6Dist.java.

◆ density() [2/2]

double umontreal.ssj.probdist.Pearson6Dist.density ( double x )

Returns \(f(x)\), the density evaluated at \(x\).

Parameters

x	value at which the density is evaluated

Returns: density function evaluated at x

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 103 of file Pearson6Dist.java.

◆ getAlpha1()

double umontreal.ssj.probdist.Pearson6Dist.getAlpha1 ( )

Returns the \(\alpha_1\) parameter of this object.

Definition at line 331 of file Pearson6Dist.java.

◆ getAlpha2()

double umontreal.ssj.probdist.Pearson6Dist.getAlpha2 ( )

Returns the \(\alpha_2\) parameter of this object.

Definition at line 338 of file Pearson6Dist.java.

◆ getBeta()

double umontreal.ssj.probdist.Pearson6Dist.getBeta ( )

Returns the \(\beta\) parameter of this object.

Definition at line 345 of file Pearson6Dist.java.

◆ getInstanceFromMLE()

Pearson6Dist umontreal.ssj.probdist.Pearson6Dist.getInstanceFromMLE	(	double[]	x,
		int	n )

static

Creates a new instance of a Pearson VI distribution with parameters.

\(\alpha_1\), \(\alpha_2\) and \(\beta\), 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

Definition at line 280 of file Pearson6Dist.java.

◆ getMean() [1/2]

double umontreal.ssj.probdist.Pearson6Dist.getMean ( )

Returns the mean.

Returns: the mean

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 122 of file Pearson6Dist.java.

◆ getMean() [2/2]

double umontreal.ssj.probdist.Pearson6Dist.getMean	(	double	alpha1,
		double	alpha2,
		double	beta )

static

Computes and returns the mean \(E[X] = (\beta\alpha_1) / (\alpha_2 - 1)\) of a Pearson VI distribution with shape parameters \(\alpha_1\) and \(\alpha_2\), and scale parameter \(\beta\).

Definition at line 290 of file Pearson6Dist.java.

◆ getMLE()

double[] umontreal.ssj.probdist.Pearson6Dist.getMLE	(	double[]	x,
		int	n )

static

Estimates the parameters \((\alpha_1,\alpha_2,\beta)\) 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: [ \(\alpha_1, \alpha_2\), \(\beta\)]. The estimate of the parameters is given by maximizing numerically the log-likelihood function, using the Uncmin package [203], [224] .

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 [ \(\hat{\alpha_1}, \hat{\alpha_2}, \hat{\beta}\)]

Definition at line 223 of file Pearson6Dist.java.

◆ getParams()

double[] umontreal.ssj.probdist.Pearson6Dist.getParams ( )

Return a table containing the parameters of the current distribution.

This table is put in regular order: [ \(\alpha_1\), \(\alpha_2\), \(\beta\)].

Implements umontreal.ssj.probdist.Distribution.

Definition at line 372 of file Pearson6Dist.java.

◆ getStandardDeviation() [1/2]

double umontreal.ssj.probdist.Pearson6Dist.getStandardDeviation ( )

Returns the standard deviation.

Returns: the standard deviation

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 130 of file Pearson6Dist.java.

◆ getStandardDeviation() [2/2]

double umontreal.ssj.probdist.Pearson6Dist.getStandardDeviation	(	double	alpha1,
		double	alpha2,
		double	beta )

static

Computes and returns the standard deviation of a Pearson VI distribution with shape parameters \(\alpha_1\) and.

\(\alpha_2\), and scale parameter \(\beta\).

Definition at line 324 of file Pearson6Dist.java.

◆ getVariance() [1/2]

double umontreal.ssj.probdist.Pearson6Dist.getVariance ( )

Returns the variance.

Returns: the variance

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 126 of file Pearson6Dist.java.

◆ getVariance() [2/2]

double umontreal.ssj.probdist.Pearson6Dist.getVariance	(	double	alpha1,
		double	alpha2,
		double	beta )

static

Computes and returns the variance \(\mbox{Var}[X] = [\beta^2 \alpha_1 (\alpha_1 + \alpha_2 - 1)] / [(\alpha_2 - 1)^2(\alpha_2 - 2)]\) of a Pearson VI distribution with shape parameters \(\alpha_1\) and \(\alpha_2\), and scale parameter \(\beta\).

Definition at line 307 of file Pearson6Dist.java.

◆ inverseF() [1/2]

double umontreal.ssj.probdist.Pearson6Dist.inverseF	(	double	alpha1,
		double	alpha2,
		double	beta,
		double	u )

static

Computes the inverse distribution function of a Pearson VI distribution with shape parameters \(\alpha_1\) and.

\(\alpha_2\), and scale parameter \(\beta\).

Definition at line 196 of file Pearson6Dist.java.

◆ inverseF() [2/2]

double umontreal.ssj.probdist.Pearson6Dist.inverseF ( double u )

Returns the inverse distribution function \(x = F^{-1}(u)\).

Restrictions: \(u \in[0,1]\).

Parameters

u	value at which the inverse distribution function is evaluated

Returns: the inverse distribution function evaluated at u

Exceptions

IllegalArgumentException if \(u\) is not in the interval \([0,1]\)

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 118 of file Pearson6Dist.java.

◆ setParam()

void umontreal.ssj.probdist.Pearson6Dist.setParam	(	double	alpha1,
		double	alpha2,
		double	beta )

Sets the parameters \(\alpha_1\), \(\alpha_2\) and.

\(\beta\) of this object.

Definition at line 354 of file Pearson6Dist.java.

◆ toString()

String umontreal.ssj.probdist.Pearson6Dist.toString ( )

Returns a String containing information about the current distribution.

Definition at line 380 of file Pearson6Dist.java.

The documentation for this class was generated from the following file:

src/main/java/umontreal/ssj/probdist/Pearson6Dist.java

SSJ API Documentation Stochastic Simulation in Java

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ Pearson6Dist()

Member Function Documentation

◆ barF() [1/2]

◆ barF() [2/2]

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ density() [1/2]

◆ density() [2/2]

◆ getAlpha1()

◆ getAlpha2()

◆ getBeta()

◆ getInstanceFromMLE()

◆ getMean() [1/2]

◆ getMean() [2/2]

◆ getMLE()

◆ getParams()

◆ getStandardDeviation() [1/2]

◆ getStandardDeviation() [2/2]

◆ getVariance() [1/2]

◆ getVariance() [2/2]

◆ inverseF() [1/2]

◆ inverseF() [2/2]

◆ setParam()

◆ toString()