Extends the class ContinuousDistribution for the inverse gamma distribution with shape parameter \(\alpha> 0\) and scale parameter. More...

Inheritance diagram for umontreal.ssj.probdist.InverseGammaDist:

Public Member Functions
	InverseGammaDist (double alpha, double beta)
	Constructs an InverseGammaDist object with parameters.
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 ()
	Returns 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 the inverse gamma distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).
static double	cdf (double alpha, double beta, double x)
	Computes the cumulative probability function of the inverse gamma distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).
static double	barF (double alpha, double beta, double x)
	Computes the complementary distribution function of the inverse gamma distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).
static double	inverseF (double alpha, double beta, double u)
	Computes the inverse distribution function of the inverse gamma distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).
static double[]	getMLE (double[] x, int n)
	Estimates the parameters \((\alpha,\beta)\) of the inverse gamma distribution using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\).
static InverseGammaDist	getInstanceFromMLE (double[] x, int n)
	Creates a new instance of the inverse gamma distribution with parameters \(\alpha\) and \(\beta\) estimated using the maximum likelihood method based on the \(n\) observations.
static double	getMean (double alpha, double beta)
	Returns the mean \(E[X] = \beta/ (\alpha- 1)\) of the inverse gamma distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).
static double	getVariance (double alpha, double beta)
	Returns the variance \(\mbox{Var}[X] = \beta^2 / ((\alpha- 1)^2(\alpha- 2))\) of the inverse gamma distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).
static double	getStandardDeviation (double alpha, double beta)
	Returns the standard deviation of the inverse gamma distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).

Detailed Description

Extends the class ContinuousDistribution for the inverse gamma distribution with shape parameter \(\alpha> 0\) and scale parameter.

\(\beta> 0\), also known as the Pearson type V distribution. The density function is given by

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

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

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

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 51 of file InverseGammaDist.java.

Constructor & Destructor Documentation

◆ InverseGammaDist()

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

Constructs an InverseGammaDist object with parameters.

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

Definition at line 61 of file InverseGammaDist.java.

Member Function Documentation

◆ barF() [1/2]

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

static

Computes the complementary distribution function of the inverse gamma distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).

Definition at line 131 of file InverseGammaDist.java.

◆ barF() [2/2]

double umontreal.ssj.probdist.InverseGammaDist.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 75 of file InverseGammaDist.java.

◆ cdf() [1/2]

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

static

Computes the cumulative probability function of the inverse gamma distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).

Definition at line 115 of file InverseGammaDist.java.

◆ cdf() [2/2]

double umontreal.ssj.probdist.InverseGammaDist.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 71 of file InverseGammaDist.java.

◆ density() [1/2]

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

static

Computes the density function of the inverse gamma distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).

Definition at line 99 of file InverseGammaDist.java.

◆ density() [2/2]

double umontreal.ssj.probdist.InverseGammaDist.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 65 of file InverseGammaDist.java.

◆ getAlpha()

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

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

Definition at line 233 of file InverseGammaDist.java.

◆ getBeta()

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

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

Definition at line 240 of file InverseGammaDist.java.

◆ getInstanceFromMLE()

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

static

Creates a new instance of the inverse gamma 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 189 of file InverseGammaDist.java.

◆ getMean() [1/2]

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

Returns the mean.

Returns: the mean

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 83 of file InverseGammaDist.java.

◆ getMean() [2/2]

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

static

Returns the mean \(E[X] = \beta/ (\alpha- 1)\) of the inverse gamma distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).

Definition at line 199 of file InverseGammaDist.java.

◆ getMLE()

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

static

Estimates the parameters \((\alpha,\beta)\) of the inverse gamma 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 167 of file InverseGammaDist.java.

◆ getParams()

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

Returns 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 262 of file InverseGammaDist.java.

◆ getStandardDeviation() [1/2]

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

Returns the standard deviation.

Returns: the standard deviation

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 91 of file InverseGammaDist.java.

◆ getStandardDeviation() [2/2]

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

static

Returns the standard deviation of the inverse gamma distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).

Definition at line 226 of file InverseGammaDist.java.

◆ getVariance() [1/2]

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

Returns the variance.

Returns: the variance

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 87 of file InverseGammaDist.java.

◆ getVariance() [2/2]

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

static

Returns the variance \(\mbox{Var}[X] = \beta^2 / ((\alpha- 1)^2(\alpha- 2))\) of the inverse gamma distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).

Definition at line 213 of file InverseGammaDist.java.

◆ inverseF() [1/2]

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

static

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

Definition at line 146 of file InverseGammaDist.java.

◆ inverseF() [2/2]

double umontreal.ssj.probdist.InverseGammaDist.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 79 of file InverseGammaDist.java.

◆ setParam()

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

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

Definition at line 247 of file InverseGammaDist.java.

◆ toString()

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

Returns a String containing information about the current distribution.

Definition at line 270 of file InverseGammaDist.java.

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

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

SSJ API Documentation Stochastic Simulation in Java

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ InverseGammaDist()

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()