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

Inheritance diagram for InverseGaussianDist:

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Collaboration diagram for InverseGaussianDist:

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Public Member Functions
	InverseGaussianDist (double mu, double lambda)
	Constructs the inverse Gaussian distribution with parameters \(\mu\) and \(\lambda\).

double	density (double x)

double	cdf (double x)
	Returns the distribution function \(F(x)\). More...

double	barF (double x)
	Returns \(\bar{F}(x) = 1 - F(x)\). More...

double	inverseF (double u)
	Returns the inverse distribution function \(F^{-1}(u)\), defined in ( inverseF ). More...

double	getMean ()
	Returns the mean of the distribution function.

double	getVariance ()
	Returns the variance of the distribution function.

double	getStandardDeviation ()
	Returns the standard deviation of the distribution function.

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. More...

String	toString ()
	Returns a `String` containing information about the current distribution.

Public Member Functions inherited from ContinuousDistribution
abstract double	density (double x)
	Returns \(f(x)\), the density evaluated at \(x\). More...

double	barF (double x)
	Returns the complementary distribution function. More...

double	inverseBrent (double a, double b, double u, double tol)
	Computes the inverse distribution function \(x = F^{-1}(u)\), using the Brent-Dekker method. More...

double	inverseBisection (double u)
	Computes and returns the inverse distribution function \(x = F^{-1}(u)\), using bisection. More...

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

double	getMean ()
	Returns the mean. More...

double	getVariance ()
	Returns the variance. More...

double	getStandardDeviation ()
	Returns the standard deviation. More...

double	getXinf ()
	Returns \(x_a\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...

double	getXsup ()
	Returns \(x_b\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...

void	setXinf (double xa)
	Sets the value \(x_a=\) `xa`, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...

void	setXsup (double xb)
	Sets the value \(x_b=\) `xb`, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...

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

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 \(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\). More...

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 \(x[i]\), \(i = 0, 1, …, n-1\). More...

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

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 \(\lambda\). More...

static double	getStandardDeviation (double mu, double lambda)
	Computes and returns the standard deviation of the inverse gaussian distribution with parameters \(\mu\) and \(\lambda\). More...

Protected Attributes
double	mu

double	lambda

Protected Attributes inherited from ContinuousDistribution
double	supportA = Double.NEGATIVE_INFINITY

double	supportB = Double.POSITIVE_INFINITY

Additional Inherited Members
Public Attributes inherited from ContinuousDistribution
int	decPrec = 15

Static Protected Attributes inherited from ContinuousDistribution
static final double	XBIG = 100.0

static final double	XBIGM = 1000.0

static final double []	EPSARRAY

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.

Member Function Documentation

◆ barF()

double barF ( double x )

Returns \(\bar{F}(x) = 1 - F(x)\).

Parameters

x	value at which the complementary distribution function is evaluated

Returns: complementary distribution function evaluated at x

Implements Distribution.

◆ cdf()

double 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 Distribution.

◆ getInstanceFromMLE()

static 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

◆ getMean()

static double 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 \(E[X] = \mu\)

◆ getMLE()

static double [] 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 \(x[0],…,x[n-1]\), [99] (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 [ \(\hat{\mu}\), \(\hat{\lambda}\)]

◆ getParams()

double [] getParams ( )

Return a table containing the parameters of the current distribution.

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

Implements Distribution.

◆ getStandardDeviation()

static double 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

◆ getVariance()

static double 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 \(\mbox{Var}[X] = \mu^3 / \lambda\)

◆ inverseF()

double inverseF ( double u )

Returns the inverse distribution function \(F^{-1}(u)\), defined in ( inverseF ).

Parameters

u	value in the interval \((0,1)\) for which the inverse distribution function is evaluated

Returns: the inverse distribution function evaluated at u

Implements Distribution.

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

InverseGaussianDist.java

Public Member Functions

Static Public Member Functions

Protected Attributes

Additional Inherited Members

Detailed Description

Member Function Documentation

◆ barF()

◆ cdf()

◆ getInstanceFromMLE()

◆ getMean()

◆ getMLE()

◆ getParams()

◆ getStandardDeviation()

◆ getVariance()

◆ inverseF()