Extends the class ContinuousDistribution for the lognormal distribution [95] . More...

Inheritance diagram for umontreal.ssj.probdist.LognormalDist:

Public Member Functions
	LognormalDist ()
	Constructs a LognormalDist object with default parameters \(\mu= 0\) and \(\sigma= 1\).
	LognormalDist (double mu, double sigma)
	Constructs a LognormalDist object with parameters \(\mu\) = mu and \(\sigma\) = sigma.
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	getMu ()
	Returns the parameter \(\mu\) of this object.
double	getSigma ()
	Returns the parameter \(\sigma\) of this object.
void	setParams (double mu, double sigma)
	Sets the parameters \(\mu\) and \(\sigma\) of this object.
double[]	getParams ()
	Returns a table containing the parameters of the current distribution, in the order: [ \(\mu\), \(\sigma\)].
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 sigma, double x)
	Computes the lognormal density function \(f(x)\) in ( `flognormal` ).
static double	cdf (double mu, double sigma, double x)
	Computes the lognormal distribution function, using NormalDist.cdf01.
static double	barF (double mu, double sigma, double x)
	Computes the lognormal complementary distribution function.
static double	inverseF (double mu, double sigma, double u)
	Computes the inverse of the lognormal distribution function, using NormalDist.inverseF01.
static double[]	getMLE (double[] x, int n)
	Estimates the parameters \((\mu, \sigma)\) of the lognormal distribution using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\).
static LognormalDist	getInstanceFromMLE (double[] x, int n)
	Creates a new instance of a lognormal distribution with parameters.
static double	getMean (double mu, double sigma)
	Computes and returns the mean \(E[X] = e^{\mu+ \sigma^2/2}\) of the lognormal distribution with parameters \(\mu\) and.
static double	getVariance (double mu, double sigma)
	Computes and returns the variance \(\mbox{Var}[X] = e^{2\mu+ \sigma^2}(e^{\sigma^2} - 1)\) of the lognormal distribution with parameters \(\mu\) and \(\sigma\).
static double	getStandardDeviation (double mu, double sigma)
	Computes and returns the standard deviation of the lognormal distribution with parameters \(\mu\) and \(\sigma\).

Detailed Description

Extends the class ContinuousDistribution for the lognormal distribution [95] .

(See also the Johnson

\(S_L\) distribution JohnsonSLDist in this package.) It has scale parameter \(\mu\) and shape parameter \(\sigma> 0\). The density is

\[ f(x) = \frac{1}{\sqrt{2\pi}\sigma x} e^{-(\ln(x) - \mu)^2/(2\sigma^2)} \qquad\mbox{for } x>0, \tag{flognormal} \]

and 0 elsewhere. The distribution function is

\[ F(x) = \Phi\left({(\ln(x) - \mu)/\sigma}\right) \qquad\mbox{for } x>0, \]

where \(\Phi\) is the standard normal distribution function. Its inverse is given by

\[ F^{-1}(u) = e^{\mu+ \sigma\Phi^{-1} (u)} \qquad\mbox{for } 0 \le u < 1. \]

If \(\ln(Y)\) has a normal distribution, then \(Y\) has a lognormal distribution with the same parameters.

This class relies on the methods NormalDist.cdf01 and NormalDist.inverseF01 of NormalDist to approximate \(\Phi\) and \(\Phi^{-1}\).

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Definition at line 55 of file LognormalDist.java.

Constructor & Destructor Documentation

◆ LognormalDist() [1/2]

umontreal.ssj.probdist.LognormalDist.LognormalDist ( )

Constructs a LognormalDist object with default parameters \(\mu= 0\) and \(\sigma= 1\).

Definition at line 63 of file LognormalDist.java.

◆ LognormalDist() [2/2]

umontreal.ssj.probdist.LognormalDist.LognormalDist	(	double	mu,
		double	sigma )

Constructs a LognormalDist object with parameters \(\mu\) = mu and \(\sigma\) = sigma.

Definition at line 71 of file LognormalDist.java.

Member Function Documentation

◆ barF() [1/2]

double umontreal.ssj.probdist.LognormalDist.barF	(	double	mu,
		double	sigma,
		double	x )

static

Computes the lognormal complementary distribution function.

\(\bar{F}(x)\), using NormalDist.barF01.

Definition at line 132 of file LognormalDist.java.

◆ barF() [2/2]

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

◆ cdf() [1/2]

double umontreal.ssj.probdist.LognormalDist.cdf	(	double	mu,
		double	sigma,
		double	x )

static

Computes the lognormal distribution function, using NormalDist.cdf01.

Definition at line 119 of file LognormalDist.java.

◆ cdf() [2/2]

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

◆ density() [1/2]

double umontreal.ssj.probdist.LognormalDist.density	(	double	mu,
		double	sigma,
		double	x )

static

Computes the lognormal density function \(f(x)\) in ( flognormal ).

Definition at line 107 of file LognormalDist.java.

◆ density() [2/2]

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

◆ getInstanceFromMLE()

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

static

Creates a new instance of a lognormal distribution with parameters.

\(\mu\) and \(\sigma\) 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 225 of file LognormalDist.java.

◆ getMean() [1/2]

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

Returns the mean.

Returns: the mean

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 91 of file LognormalDist.java.

◆ getMean() [2/2]

double umontreal.ssj.probdist.LognormalDist.getMean	(	double	mu,
		double	sigma )

static

Computes and returns the mean \(E[X] = e^{\mu+ \sigma^2/2}\) of the lognormal distribution with parameters \(\mu\) and.

\(\sigma\).

Returns: the mean of the lognormal distribution

Definition at line 236 of file LognormalDist.java.

◆ getMLE()

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

static

Estimates the parameters \((\mu, \sigma)\) of the lognormal 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: [ \(\mu\), \(\sigma\)]. The maximum likelihood estimators are the values

\((\hat{\mu}, \hat{\sigma})\) that satisfy the equations:

\begin{align*} \hat{\mu} & = \frac{1}{n} \sum_{i=1}^n \ln(x_i) \\ \hat{\sigma} & = \sqrt{\frac{1}{n} \sum_{i=1}^n (\ln(x_i) - \hat{\mu})^2}. \end{align*}

See [95] (page 220).

Parameters

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

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

Definition at line 186 of file LognormalDist.java.

◆ getMu()

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

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

Definition at line 270 of file LognormalDist.java.

◆ getParams()

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

Returns a table containing the parameters of the current distribution, in the order: [ \(\mu\), \(\sigma\)].

Implements umontreal.ssj.probdist.Distribution.

Definition at line 296 of file LognormalDist.java.

◆ getSigma()

double umontreal.ssj.probdist.LognormalDist.getSigma ( )

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

Definition at line 277 of file LognormalDist.java.

◆ getStandardDeviation() [1/2]

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

Returns the standard deviation.

Returns: the standard deviation

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 99 of file LognormalDist.java.

◆ getStandardDeviation() [2/2]

double umontreal.ssj.probdist.LognormalDist.getStandardDeviation	(	double	mu,
		double	sigma )

static

Computes and returns the standard deviation of the lognormal distribution with parameters \(\mu\) and \(\sigma\).

Returns: the standard deviation of the lognormal distribution

Definition at line 263 of file LognormalDist.java.

◆ getVariance() [1/2]

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

Returns the variance.

Returns: the variance

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 95 of file LognormalDist.java.

◆ getVariance() [2/2]

double umontreal.ssj.probdist.LognormalDist.getVariance	(	double	mu,
		double	sigma )

static

Computes and returns the variance \(\mbox{Var}[X] = e^{2\mu+ \sigma^2}(e^{\sigma^2} - 1)\) of the lognormal distribution with parameters \(\mu\) and \(\sigma\).

Returns: the variance of the lognormal distribution

Definition at line 250 of file LognormalDist.java.

◆ inverseF() [1/2]

double umontreal.ssj.probdist.LognormalDist.inverseF	(	double	mu,
		double	sigma,
		double	u )

static

Computes the inverse of the lognormal distribution function, using NormalDist.inverseF01.

Definition at line 144 of file LognormalDist.java.

◆ inverseF() [2/2]

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

◆ setParams()

void umontreal.ssj.probdist.LognormalDist.setParams	(	double	mu,
		double	sigma )

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

Definition at line 284 of file LognormalDist.java.

◆ toString()

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

Returns a String containing information about the current distribution.

Definition at line 304 of file LognormalDist.java.

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

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

SSJ API Documentation Stochastic Simulation in Java

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ LognormalDist() [1/2]

◆ LognormalDist() [2/2]

Member Function Documentation

◆ barF() [1/2]

◆ barF() [2/2]

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ density() [1/2]

◆ density() [2/2]

◆ getInstanceFromMLE()

◆ getMean() [1/2]

◆ getMean() [2/2]

◆ getMLE()

◆ getMu()

◆ getParams()

◆ getSigma()

◆ getStandardDeviation() [1/2]

◆ getStandardDeviation() [2/2]

◆ getVariance() [1/2]

◆ getVariance() [2/2]

◆ inverseF() [1/2]

◆ inverseF() [2/2]

◆ setParams()

◆ toString()