Extends the class ContinuousDistribution for the Log-Logistic distribution with shape parameter \(\alpha> 0\) and scale parameter. More...

Inheritance diagram for umontreal.ssj.probdist.LoglogisticDist:

Public Member Functions
	LoglogisticDist (double alpha, double beta)
	Constructs a log-logistic distribution with parameters \(\alpha\) and \(\beta\).
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	getAlpha ()
	Return the parameter \(\alpha\) of this object.
double	getBeta ()
	Returns the parameter \(\beta\) of this object.
void	setParams (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 ( `floglogistic` ) for a log-logisitic distribution with parameters.
static double	cdf (double alpha, double beta, double x)
	Computes the distribution function ( `Floglogistic` ) of the log-logistic distribution with parameters.
static double	barF (double alpha, double beta, double x)
	Computes the complementary distribution function ( `Fbarloglogistic` ) of the log-logistic distribution with parameters.
static double	inverseF (double alpha, double beta, double u)
	Computes the inverse of the log-logistic distribution with parameters \(\alpha\) and \(\beta\).
static double[]	getMLE (double[] x, int n)
	Estimates the parameters \((\alpha,\beta)\) of the log-logistic distribution using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\).
static LoglogisticDist	getInstanceFromMLE (double[] x, int n)
	Creates a new instance of a log-logistic distribution with parameters \(\alpha\) and \(\beta\) estimated using the maximum likelihood method based on the \(n\) observations.
static double	getMean (double alpha, double beta)
	Computes and returns the mean \(E[X] = \beta\theta \mbox{cosec}(\theta), \mbox{ where } \theta= \pi/\alpha,\) of the log-logistic distribution with parameters \(\alpha\) and.
static double	getVariance (double alpha, double beta)
	Computes and returns the variance.
static double	getStandardDeviation (double alpha, double beta)
	Computes and returns the standard deviation of the log-logistic distribution with parameters \(\alpha\) and \(\beta\).

Detailed Description

Extends the class ContinuousDistribution for the Log-Logistic distribution with shape parameter \(\alpha> 0\) and scale parameter.

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

\[ f(x) = \frac{\alpha(x / \beta)^{\alpha- 1}}{\beta[1 + (x / \beta)^{\alpha}]^2} \qquad\qquad\mbox{for } x > 0 \tag{floglogistic} \]

and its distribution function is

\[ F(x) = \frac{1}{1 + (\frac{x}{\beta})^{-\alpha}} \qquad\qquad\mbox{for } x > 0. \tag{Floglogistic} \]

The complementary distribution is

\[ \bar{F}(x) = \frac{1}{1 + (\frac{x}{\beta})^{\alpha}} \qquad\qquad\mbox{for } x > 0. \tag{Fbarloglogistic} \]

Definition at line 49 of file LoglogisticDist.java.

Constructor & Destructor Documentation

◆ LoglogisticDist()

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

Constructs a log-logistic distribution with parameters \(\alpha\) and \(\beta\).

Definition at line 89 of file LoglogisticDist.java.

Member Function Documentation

◆ barF() [1/2]

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

static

Computes the complementary distribution function ( Fbarloglogistic ) of the log-logistic distribution with parameters.

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

Definition at line 181 of file LoglogisticDist.java.

◆ barF() [2/2]

double umontreal.ssj.probdist.LoglogisticDist.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 101 of file LoglogisticDist.java.

◆ cdf() [1/2]

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

static

Computes the distribution function ( Floglogistic ) of the log-logistic distribution with parameters.

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

Definition at line 158 of file LoglogisticDist.java.

◆ cdf() [2/2]

double umontreal.ssj.probdist.LoglogisticDist.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 97 of file LoglogisticDist.java.

◆ density() [1/2]

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

static

Computes the density function ( floglogistic ) for a log-logisitic distribution with parameters.

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

Definition at line 128 of file LoglogisticDist.java.

◆ density() [2/2]

double umontreal.ssj.probdist.LoglogisticDist.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 93 of file LoglogisticDist.java.

◆ getAlpha()

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

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

Definition at line 352 of file LoglogisticDist.java.

◆ getBeta()

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

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

Definition at line 359 of file LoglogisticDist.java.

◆ getInstanceFromMLE()

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

static

Creates a new instance of a log-logistic 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 287 of file LoglogisticDist.java.

◆ getMean() [1/2]

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

Returns the mean.

Returns: the mean

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 109 of file LoglogisticDist.java.

◆ getMean() [2/2]

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

static

Computes and returns the mean \(E[X] = \beta\theta \mbox{cosec}(\theta), \mbox{ where } \theta= \pi/\alpha,\) of the log-logistic distribution with parameters \(\alpha\) and.

\(\beta\).

Returns: the mean of the log-logistic distribution \(E[X] = \beta\theta \mbox{cosec}(\theta), \mbox{ where } \theta= \pi/ \alpha\)

Definition at line 301 of file LoglogisticDist.java.

◆ getMLE()

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

static

Estimates the parameters \((\alpha,\beta)\) of the log-logistic 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 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 [ \(\hat{\alpha}\), \(\hat{\beta}\)]

Definition at line 236 of file LoglogisticDist.java.

◆ getParams()

double[] umontreal.ssj.probdist.LoglogisticDist.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 381 of file LoglogisticDist.java.

◆ getStandardDeviation() [1/2]

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

Returns the standard deviation.

Returns: the standard deviation

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 117 of file LoglogisticDist.java.

◆ getStandardDeviation() [2/2]

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

static

Computes and returns the standard deviation of the log-logistic distribution with parameters \(\alpha\) and \(\beta\).

Returns: the standard deviation of the log-logistic distribution

Definition at line 345 of file LoglogisticDist.java.

◆ getVariance() [1/2]

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

Returns the variance.

Returns: the variance

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 113 of file LoglogisticDist.java.

◆ getVariance() [2/2]

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

static

Computes and returns the variance.

\(\mbox{Var}[X] = \beta^2 \theta(2 \mbox{cosec}(2 \theta) - \theta[\mbox{cosec}(\theta)]^2), \mbox{ where } \theta= {\pi/\alpha},\) of the log-logistic distribution with parameters \(\alpha\) and \(\beta\).

Returns: the variance of the log-logistic distribution \(\mbox{Var}[X] = \beta^2 \theta(2 \mbox{cosec}(2 \theta) - \theta[\mbox{cosec}(\theta)]^2), \mbox{ where } \theta= \pi/ \alpha\)

Definition at line 326 of file LoglogisticDist.java.

◆ inverseF() [1/2]

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

static

Computes the inverse of the log-logistic distribution with parameters \(\alpha\) and \(\beta\).

Definition at line 204 of file LoglogisticDist.java.

◆ inverseF() [2/2]

double umontreal.ssj.probdist.LoglogisticDist.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 105 of file LoglogisticDist.java.

◆ setParams()

void umontreal.ssj.probdist.LoglogisticDist.setParams	(	double	alpha,
		double	beta )

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

Definition at line 366 of file LoglogisticDist.java.

◆ toString()

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

Returns a String containing information about the current distribution.

Definition at line 389 of file LoglogisticDist.java.

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

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

SSJ API Documentation Stochastic Simulation in Java

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ LoglogisticDist()

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]

◆ setParams()

◆ toString()