THIS CLASS HAS BEEN RENAMED InverseGammaDist . More...

Inheritance diagram for umontreal.ssj.probdist.Pearson5Dist:

Public Member Functions
	Pearson5Dist (double alpha, double beta)
	THIS CLASS HAS BEEN RENAMED InverseGammaDist .
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	getAlpha ()
	Returns the \(\alpha\) parameter of this object.
double	getBeta ()
	Returns the \(\beta\) parameter of this object.
void	setParam (double alpha, double beta)
	Sets the parameters \(\alpha\) and \(\beta\) of this object.
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 alpha, double beta, double x)
	Computes the density function of a Pearson V distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).
static double	cdf (double alpha, double beta, double x)
	Computes the density function of a Pearson V distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).
static double	barF (double alpha, double beta, double x)
	Computes the complementary distribution function of a Pearson V distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).
static double	inverseF (double alpha, double beta, double u)
	Computes the inverse distribution function of a Pearson V distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).
static double[]	getMLE (double[] x, int n)
	Estimates the parameters \((\alpha,\beta)\) of the Pearson V distribution using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\).
static Pearson5Dist	getInstanceFromMLE (double[] x, int n)
	Creates a new instance of a Pearson V distribution with parameters.
static double	getMean (double alpha, double beta)
	Computes and returns the mean \(E[X] = \beta/ (\alpha- 1)\) of a Pearson V distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).
static double	getVariance (double alpha, double beta)
	Computes and returns the variance \(\mbox{Var}[X] = \beta^2 / ((\alpha- 1)^2(\alpha- 2)\) of a Pearson V distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).
static double	getStandardDeviation (double alpha, double beta)
	Computes and returns the standard deviation of a Pearson V distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).

Detailed Description

THIS CLASS HAS BEEN RENAMED InverseGammaDist .

Extends the class ContinuousDistribution for the Pearson type V distribution with shape parameter \(\alpha> 0\) and scale parameter

\(\beta> 0\). The density function is given by

\[ f(x) = \left\{\begin{array}{ll} \displaystyle\frac{\beta^{\alpha}e^{-\beta/ x}}{x^{\alpha+ 1} \Gamma(\alpha)} & \quad\mbox{for } x > 0 \\ 0 & \quad\mbox{otherwise,} \end{array} \right. \tag{fpearson5} \]

where \(\Gamma\) is the gamma function. The distribution function is given by

\[ F(x) = 1 - F_G\left(\frac{1}{x}\right) \qquad\mbox{for } x > 0, \tag{Fpearson5} \]

and \(F(x) = 0\) otherwise, where \(F_G(x)\) is the distribution function of a gamma distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).

Definition at line 53 of file Pearson5Dist.java.

Constructor & Destructor Documentation

◆ Pearson5Dist()

umontreal.ssj.probdist.Pearson5Dist.Pearson5Dist	(	double	alpha,
		double	beta )

THIS CLASS HAS BEEN RENAMED InverseGammaDist .

Constructs a Pearson5Dist object with parameters

\(\alpha\) = alpha and \(\beta\) = beta.

Definition at line 64 of file Pearson5Dist.java.

Member Function Documentation

◆ barF() [1/2]

double umontreal.ssj.probdist.Pearson5Dist.barF	(	double	alpha,
		double	beta,
		double	x )

static

Computes the complementary distribution function of a Pearson V distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).

Definition at line 132 of file Pearson5Dist.java.

◆ barF() [2/2]

double umontreal.ssj.probdist.Pearson5Dist.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 78 of file Pearson5Dist.java.

◆ cdf() [1/2]

double umontreal.ssj.probdist.Pearson5Dist.cdf	(	double	alpha,
		double	beta,
		double	x )

static

Computes the density function of a Pearson V distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).

Definition at line 117 of file Pearson5Dist.java.

◆ cdf() [2/2]

double umontreal.ssj.probdist.Pearson5Dist.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 74 of file Pearson5Dist.java.

◆ density() [1/2]

double umontreal.ssj.probdist.Pearson5Dist.density	(	double	alpha,
		double	beta,
		double	x )

static

Computes the density function of a Pearson V distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).

Definition at line 102 of file Pearson5Dist.java.

◆ density() [2/2]

double umontreal.ssj.probdist.Pearson5Dist.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 68 of file Pearson5Dist.java.

◆ getAlpha()

double umontreal.ssj.probdist.Pearson5Dist.getAlpha ( )

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

Definition at line 234 of file Pearson5Dist.java.

◆ getBeta()

double umontreal.ssj.probdist.Pearson5Dist.getBeta ( )

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

Definition at line 241 of file Pearson5Dist.java.

◆ getInstanceFromMLE()

Pearson5Dist umontreal.ssj.probdist.Pearson5Dist.getInstanceFromMLE	(	double[]	x,
		int	n )

static

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

\(\alpha\) 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 190 of file Pearson5Dist.java.

◆ getMean() [1/2]

double umontreal.ssj.probdist.Pearson5Dist.getMean ( )

Returns the mean.

Returns: the mean

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 86 of file Pearson5Dist.java.

◆ getMean() [2/2]

double umontreal.ssj.probdist.Pearson5Dist.getMean	(	double	alpha,
		double	beta )

static

Computes and returns the mean \(E[X] = \beta/ (\alpha- 1)\) of a Pearson V distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).

Definition at line 200 of file Pearson5Dist.java.

◆ getMLE()

double[] umontreal.ssj.probdist.Pearson5Dist.getMLE	(	double[]	x,
		int	n )

static

Estimates the parameters \((\alpha,\beta)\) of the Pearson V distribution using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\).

The estimates are returned in a two-element array, in regular order: [ \(\alpha\), \(\beta\)]. The equations of the maximum likelihood are the same as the equations of the gamma distribution, with the sample \(y_i = 1/x_i\).

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 [ \(\hat{\alpha}, \hat{\beta}\)]

Definition at line 168 of file Pearson5Dist.java.

◆ getParams()

double[] umontreal.ssj.probdist.Pearson5Dist.getParams ( )

Return a table containing the parameters of the current distribution.

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

Implements umontreal.ssj.probdist.Distribution.

Definition at line 263 of file Pearson5Dist.java.

◆ getStandardDeviation() [1/2]

double umontreal.ssj.probdist.Pearson5Dist.getStandardDeviation ( )

Returns the standard deviation.

Returns: the standard deviation

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 94 of file Pearson5Dist.java.

◆ getStandardDeviation() [2/2]

double umontreal.ssj.probdist.Pearson5Dist.getStandardDeviation	(	double	alpha,
		double	beta )

static

Computes and returns the standard deviation of a Pearson V distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).

Definition at line 227 of file Pearson5Dist.java.

◆ getVariance() [1/2]

double umontreal.ssj.probdist.Pearson5Dist.getVariance ( )

Returns the variance.

Returns: the variance

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 90 of file Pearson5Dist.java.

◆ getVariance() [2/2]

double umontreal.ssj.probdist.Pearson5Dist.getVariance	(	double	alpha,
		double	beta )

static

Computes and returns the variance \(\mbox{Var}[X] = \beta^2 / ((\alpha- 1)^2(\alpha- 2)\) of a Pearson V distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).

Definition at line 214 of file Pearson5Dist.java.

◆ inverseF() [1/2]

double umontreal.ssj.probdist.Pearson5Dist.inverseF	(	double	alpha,
		double	beta,
		double	u )

static

Computes the inverse distribution function of a Pearson V distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).

Definition at line 147 of file Pearson5Dist.java.

◆ inverseF() [2/2]

double umontreal.ssj.probdist.Pearson5Dist.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 82 of file Pearson5Dist.java.

◆ setParam()

void umontreal.ssj.probdist.Pearson5Dist.setParam	(	double	alpha,
		double	beta )

Sets the parameters \(\alpha\) and \(\beta\) of this object.

Definition at line 248 of file Pearson5Dist.java.

◆ toString()

String umontreal.ssj.probdist.Pearson5Dist.toString ( )

Returns a String containing information about the current distribution.

Definition at line 271 of file Pearson5Dist.java.

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

src/main/java/umontreal/ssj/probdist/Pearson5Dist.java

SSJ API Documentation Stochastic Simulation in Java

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ Pearson5Dist()

Member Function Documentation

◆ barF() [1/2]

◆ barF() [2/2]

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ density() [1/2]

◆ density() [2/2]

◆ getAlpha()

◆ 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()