Extends the class ContinuousDistribution for the half-normal distribution with parameters \(\mu\) and \(\sigma> 0\). More...

Inheritance diagram for umontreal.ssj.probdist.HalfNormalDist:

Public Member Functions
	HalfNormalDist (double mu, double sigma)
	Constructs a HalfNormalDist 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 \(\sigma\).
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 sigma, double x)
	Computes the density function of the half-normal distribution.
static double	cdf (double mu, double sigma, double x)
	Computes the distribution function.
static double	barF (double mu, double sigma, double x)
	Computes the complementary distribution function.
static double	inverseF (double mu, double sigma, double u)
	Computes the inverse of the distribution function.
static double[]	getMLE (double[] x, int n)
	Estimates the parameters \(\mu\) and \(\sigma\) of the half-normal distribution using the maximum likelihood method from the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\).
static double[]	getMLE (double[] x, int n, double mu)
	Estimates the parameter \(\sigma\) of the half-normal distribution using the maximum likelihood method from the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\) and the parameter.
static double	getMean (double mu, double sigma)
	Computes and returns the mean \( E[X] = \mu+ \sigma\sqrt{2 / \pi}. \).
static double	getVariance (double mu, double sigma)
	Computes and returns the variance \( \mbox{Var}[X] = \left(1-2/\pi\right)\sigma^2. \).
static double	getStandardDeviation (double mu, double sigma)
	Computes the standard deviation of the half-normal distribution with parameters \(\mu\) and \(\sigma\).

Detailed Description

Extends the class ContinuousDistribution for the half-normal distribution with parameters \(\mu\) and \(\sigma> 0\).

Its density is

\begin{align} f(x) & = \frac{1}{\sigma}\sqrt{\frac{2}{\pi}}\; e^{-(x-\mu)^2/2\sigma^2}, \qquad\mbox{for } x \ge\mu. \\ \tag{fHalfNormal} f(x) & = 0, \qquad\mbox{for } x < \mu. \nonumber \end{align}

Definition at line 43 of file HalfNormalDist.java.

Constructor & Destructor Documentation

◆ HalfNormalDist()

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

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

Definition at line 52 of file HalfNormalDist.java.

Member Function Documentation

◆ barF() [1/2]

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

static

Computes the complementary distribution function.

Parameters

mu	the parameter mu
sigma	the parameter sigma
x	the value at which the complementary distribution is evaluated

Returns: returns the complementary distribution function

Definition at line 129 of file HalfNormalDist.java.

◆ barF() [2/2]

double umontreal.ssj.probdist.HalfNormalDist.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 67 of file HalfNormalDist.java.

◆ cdf() [1/2]

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

static

Computes the distribution function.

Parameters

mu	the parameter mu
sigma	the parameter sigma
x	the value at which the distribution is evaluated

Returns: returns the cdf function

Definition at line 112 of file HalfNormalDist.java.

◆ cdf() [2/2]

double umontreal.ssj.probdist.HalfNormalDist.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 63 of file HalfNormalDist.java.

◆ density() [1/2]

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

static

Computes the density function of the half-normal distribution.

Parameters

mu	the parameter mu
sigma	the parameter sigma
x	the value at which the density is evaluated

Returns: returns the density function

Definition at line 95 of file HalfNormalDist.java.

◆ density() [2/2]

double umontreal.ssj.probdist.HalfNormalDist.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 56 of file HalfNormalDist.java.

◆ getMean() [1/2]

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

Returns the mean.

Returns: the mean

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 75 of file HalfNormalDist.java.

◆ getMean() [2/2]

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

static

Computes and returns the mean \( E[X] = \mu+ \sigma\sqrt{2 / \pi}. \).

Parameters

mu	the parameter mu
sigma	the parameter sigma

Returns: returns the mean

Definition at line 229 of file HalfNormalDist.java.

◆ getMLE() [1/2]

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

static

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

\(\hat{\mu}\) and \(\hat{\sigma}\) that satisfy the equation

\begin{align*} \hat{\mu}= \min_j \{x_j\}, \\ \hat{\sigma}= \sqrt{\frac{1}{n}\Sigma_j(x_j-\hat{\mu})^2}. \end{align*}

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 [ \(\mu\), \(\sigma\)]

Definition at line 175 of file HalfNormalDist.java.

◆ getMLE() [2/2]

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

static

Estimates the parameter \(\sigma\) of the half-normal distribution using the maximum likelihood method from the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\) and the parameter.

\(\mu\) = mu. The estimate is returned in a one-element array: [ \(\sigma\)]. The maximum likelihood estimator is the value \(\hat{\sigma}\) that satisfies the equation

\begin{align*} \hat{\sigma}= \sqrt{\frac{1}{n}\Sigma_j(x_j-\mu)^2}. \end{align*}

Parameters

x	the list of observations to use to evaluate parameters
n	the number of observations to use to evaluate parameter
mu	the parameter mu

Returns: returns the parameter [ \(\sigma\)]

Definition at line 209 of file HalfNormalDist.java.

◆ getMu()

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

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

Returns: returns the parameter mu

Definition at line 266 of file HalfNormalDist.java.

◆ getParams()

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

Return a table containing the parameters of the current distribution.

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

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

Implements umontreal.ssj.probdist.Distribution.

Definition at line 299 of file HalfNormalDist.java.

◆ getSigma()

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

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

Returns: returns the parameter sigma

Definition at line 275 of file HalfNormalDist.java.

◆ getStandardDeviation() [1/2]

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

Returns the standard deviation.

Returns: the standard deviation

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 83 of file HalfNormalDist.java.

◆ getStandardDeviation() [2/2]

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

static

Computes the standard deviation of the half-normal distribution with parameters \(\mu\) and \(\sigma\).

Parameters

mu	the parameter mu
sigma	the parameter sigma

Returns: returns the standard deviation

Definition at line 257 of file HalfNormalDist.java.

◆ getVariance() [1/2]

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

Returns the variance.

Returns: the variance

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 79 of file HalfNormalDist.java.

◆ getVariance() [2/2]

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

static

Computes and returns the variance \( \mbox{Var}[X] = \left(1-2/\pi\right)\sigma^2. \).

Parameters

mu	the parameter mu
sigma	the parameter sigma

Returns: returns the variance

Definition at line 243 of file HalfNormalDist.java.

◆ inverseF() [1/2]

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

static

Computes the inverse of the distribution function.

Parameters

mu	the parameter mu
sigma	the parameter sigma
u	the value at which the inverse distribution is evaluated

Returns: returns the inverse distribution function

Definition at line 146 of file HalfNormalDist.java.

◆ inverseF() [2/2]

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

◆ setParams()

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

Sets the parameters \(\mu\) and \(\sigma\).

Parameters

mu	the parameter mu
sigma	the parameter sigma

Definition at line 285 of file HalfNormalDist.java.

◆ toString()

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

Returns a String containing information about the current distribution.

Returns: returns a String containing information about the current distribution.

Definition at line 310 of file HalfNormalDist.java.

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

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

SSJ API Documentation Stochastic Simulation in Java

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ HalfNormalDist()

Member Function Documentation

◆ barF() [1/2]

◆ barF() [2/2]

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ density() [1/2]

◆ density() [2/2]

◆ getMean() [1/2]

◆ getMean() [2/2]

◆ getMLE() [1/2]

◆ getMLE() [2/2]

◆ getMu()

◆ getParams()

◆ getSigma()

◆ getStandardDeviation() [1/2]

◆ getStandardDeviation() [2/2]

◆ getVariance() [1/2]

◆ getVariance() [2/2]

◆ inverseF() [1/2]

◆ inverseF() [2/2]

◆ setParams()

◆ toString()