Extends the class ContinuousDistribution for the inverse Gaussian distribution with location parameter \(\mu> 0\) and scale parameter. More...

Inheritance diagram for umontreal.ssj.probdist.InverseGaussianDist:

Public Member Functions
	InverseGaussianDist (double mu, double lambda)
	Constructs the inverse Gaussian distribution 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	getLambda ()
	Returns the parameter \(\lambda\) of this object.
double	getMu ()
	Returns the parameter \(\mu\) of this object.
void	setParams (double mu, double lambda)
	Sets the parameters \(\mu\) and \(\lambda\) 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 mu, double lambda, double x)
	Computes the density function ( `fInverseGaussian` ) for the inverse gaussian distribution with parameters.
static double	cdf (double mu, double lambda, double x)
	Computes the distribution function ( `FInverseGaussian` ) of the inverse gaussian distribution with parameters \(\mu\) and \(\lambda\), evaluated at \(x\).
static double	barF (double mu, double lambda, double x)
	Computes the complementary distribution function of the inverse gaussian distribution with parameters \(\mu\) and \(\lambda\), evaluated at \(x\).
static double	inverseF (double mu, double lambda, double u)
	Computes the inverse of the inverse gaussian distribution with parameters \(\mu\) and \(\lambda\).
static double[]	getMLE (double[] x, int n)
	Estimates the parameters \((\mu, \lambda)\) of the inverse gaussian distribution using the maximum likelihood method, from the.
static InverseGaussianDist	getInstanceFromMLE (double[] x, int n)
	Creates a new instance of an inverse gaussian distribution with parameters \(\mu\) and \(\lambda\) estimated using the maximum likelihood method based on the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\).
static double	getMean (double mu, double lambda)
	Returns the mean \(E[X] = \mu\) of the inverse gaussian distribution with parameters \(\mu\) and \(\lambda\).
static double	getVariance (double mu, double lambda)
	Computes and returns the variance \(\mbox{Var}[X] = \mu^3/\lambda\) of the inverse gaussian distribution with parameters \(\mu\) and \(\lambda\).
static double	getStandardDeviation (double mu, double lambda)
	Computes and returns the standard deviation of the inverse gaussian distribution with parameters \(\mu\) and \(\lambda\).

Detailed Description

Extends the class ContinuousDistribution for the inverse Gaussian distribution with location parameter \(\mu> 0\) and scale parameter.

\(\lambda> 0\). Its density is

\[ f(x) = \sqrt{\frac{\lambda}{2\pi x^3}}\; e^{{-\lambda(x - \mu)^2}/{(2\mu^2x)}}, \qquad\mbox{for } x > 0. \tag{fInverseGaussian} \]

The distribution function is given by

\[ F(x) = \Phi\left(\sqrt{\frac{\lambda}{x}}\left(\frac{x}{\mu} - 1\right)\right) + e^{({2\lambda}/{\mu})}\Phi\left(-\sqrt{\frac{\lambda}{x}}\left(\frac{x}{\mu} + 1\right)\right), \tag{FInverseGaussian} \]

where \(\Phi\) is the standard normal distribution function.

The non-static versions of the methods cdf, barF, and inverseF call the static version of the same name.

Definition at line 54 of file InverseGaussianDist.java.

Constructor & Destructor Documentation

◆ InverseGaussianDist()

umontreal.ssj.probdist.InverseGaussianDist.InverseGaussianDist	(	double	mu,
		double	lambda )

Constructs the inverse Gaussian distribution with parameters.

\(\mu\) and \(\lambda\).

Definition at line 79 of file InverseGaussianDist.java.

Member Function Documentation

◆ barF() [1/2]

double umontreal.ssj.probdist.InverseGaussianDist.barF	(	double	mu,
		double	lambda,
		double	x )

static

Computes the complementary distribution function of the inverse gaussian distribution with parameters \(\mu\) and \(\lambda\), evaluated at \(x\).

Definition at line 158 of file InverseGaussianDist.java.

◆ barF() [2/2]

double umontreal.ssj.probdist.InverseGaussianDist.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 91 of file InverseGaussianDist.java.

◆ cdf() [1/2]

double umontreal.ssj.probdist.InverseGaussianDist.cdf	(	double	mu,
		double	lambda,
		double	x )

static

Computes the distribution function ( FInverseGaussian ) of the inverse gaussian distribution with parameters \(\mu\) and \(\lambda\), evaluated at \(x\).

Definition at line 138 of file InverseGaussianDist.java.

◆ cdf() [2/2]

double umontreal.ssj.probdist.InverseGaussianDist.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 87 of file InverseGaussianDist.java.

◆ density() [1/2]

double umontreal.ssj.probdist.InverseGaussianDist.density	(	double	mu,
		double	lambda,
		double	x )

static

Computes the density function ( fInverseGaussian ) for the inverse gaussian distribution with parameters.

\(\mu\) and \(\lambda\), evaluated at \(x\).

Definition at line 118 of file InverseGaussianDist.java.

◆ density() [2/2]

double umontreal.ssj.probdist.InverseGaussianDist.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 83 of file InverseGaussianDist.java.

◆ getInstanceFromMLE()

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

static

Creates a new instance of an inverse gaussian distribution with parameters \(\mu\) and \(\lambda\) 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 242 of file InverseGaussianDist.java.

◆ getLambda()

double umontreal.ssj.probdist.InverseGaussianDist.getLambda ( )

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

Definition at line 291 of file InverseGaussianDist.java.

◆ getMean() [1/2]

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

Returns the mean.

Returns: the mean

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 99 of file InverseGaussianDist.java.

◆ getMean() [2/2]

double umontreal.ssj.probdist.InverseGaussianDist.getMean	(	double	mu,
		double	lambda )

static

Returns the mean \(E[X] = \mu\) of the inverse gaussian distribution with parameters \(\mu\) and \(\lambda\).

Returns: the mean of the inverse gaussian distribution \(E[X] = \mu\)

Definition at line 253 of file InverseGaussianDist.java.

◆ getMLE()

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

static

Estimates the parameters \((\mu, \lambda)\) of the inverse gaussian 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: [ \(\mu\), \(\lambda\)]. The maximum likelihood estimators are the values \((\hat{\mu}, \hat{\lambda})\) that satisfy the equations:

\begin{align*} \hat{\mu} & = \bar{x}_n \\ \frac{1}{\hat{\lambda}} & = \frac{1}{n} \sum_{i=1}^n \left(\frac{1}{x_i} - \frac{1}{\hat{\mu}}\right), \end{align*}

where \(\bar{x}_n\) is the average of \(x[0],…,x[n-1]\), [95] (page 271).

Parameters

x	the list of observations used to evaluate parameters
n	the number of observations used to evaluate parameters

Returns: returns the parameters [ \(\hat{\mu}\), \(\hat{\lambda}\)]

Definition at line 212 of file InverseGaussianDist.java.

◆ getMu()

double umontreal.ssj.probdist.InverseGaussianDist.getMu ( )

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

Definition at line 298 of file InverseGaussianDist.java.

◆ getParams()

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

Return a table containing the parameters of the current distribution.

This table is put in regular order: [ \(\mu\), \(\lambda\)].

Implements umontreal.ssj.probdist.Distribution.

Definition at line 320 of file InverseGaussianDist.java.

◆ getStandardDeviation() [1/2]

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

Returns the standard deviation.

Returns: the standard deviation

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 107 of file InverseGaussianDist.java.

◆ getStandardDeviation() [2/2]

double umontreal.ssj.probdist.InverseGaussianDist.getStandardDeviation	(	double	mu,
		double	lambda )

static

Computes and returns the standard deviation of the inverse gaussian distribution with parameters \(\mu\) and \(\lambda\).

Returns: the standard deviation of the inverse gaussian distribution

Definition at line 284 of file InverseGaussianDist.java.

◆ getVariance() [1/2]

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

Returns the variance.

Returns: the variance

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 103 of file InverseGaussianDist.java.

◆ getVariance() [2/2]

double umontreal.ssj.probdist.InverseGaussianDist.getVariance	(	double	mu,
		double	lambda )

static

Computes and returns the variance \(\mbox{Var}[X] = \mu^3/\lambda\) of the inverse gaussian distribution with parameters \(\mu\) and \(\lambda\).

Returns: the variance of the inverse gaussian distribution \(\mbox{Var}[X] = \mu^3 / \lambda\)

Definition at line 269 of file InverseGaussianDist.java.

◆ inverseF() [1/2]

double umontreal.ssj.probdist.InverseGaussianDist.inverseF	(	double	mu,
		double	lambda,
		double	u )

static

Computes the inverse of the inverse gaussian distribution with parameters \(\mu\) and \(\lambda\).

Definition at line 166 of file InverseGaussianDist.java.

◆ inverseF() [2/2]

double umontreal.ssj.probdist.InverseGaussianDist.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 95 of file InverseGaussianDist.java.

◆ setParams()

void umontreal.ssj.probdist.InverseGaussianDist.setParams	(	double	mu,
		double	lambda )

Sets the parameters \(\mu\) and \(\lambda\) of this object.

Definition at line 305 of file InverseGaussianDist.java.

◆ toString()

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

Returns a String containing information about the current distribution.

Definition at line 328 of file InverseGaussianDist.java.

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

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

SSJ API Documentation Stochastic Simulation in Java

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ InverseGaussianDist()

Member Function Documentation

◆ barF() [1/2]

◆ barF() [2/2]

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ density() [1/2]

◆ density() [2/2]

◆ getInstanceFromMLE()

◆ getLambda()

◆ getMean() [1/2]

◆ getMean() [2/2]

◆ getMLE()

◆ getMu()

◆ getParams()

◆ getStandardDeviation() [1/2]

◆ getStandardDeviation() [2/2]

◆ getVariance() [1/2]

◆ getVariance() [2/2]

◆ inverseF() [1/2]

◆ inverseF() [2/2]

◆ setParams()

◆ toString()