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

Inheritance diagram for HalfNormalDist:

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Collaboration diagram for HalfNormalDist:

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Public Member Functions
	HalfNormalDist (double mu, double sigma)
	Constructs a `HalfNormalDist` object with parameters \(\mu=\) `mu` and \(\sigma=\) `sigma`.

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

double	getSigma ()
	Returns the parameter \(\sigma\) of this object. More...

void	setParams (double mu, double sigma)
	Sets the parameters \(\mu\) and \(\sigma\). More...

double []	getParams ()
	Return a table containing the parameters of the current distribution. More...

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

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 sigma, double x)
	Computes the density function of the half-normal distribution. More...

static double	cdf (double mu, double sigma, double x)
	Computes the distribution function. More...

static double	barF (double mu, double sigma, double x)
	Computes the complementary distribution function. More...

static double	inverseF (double mu, double sigma, double u)
	Computes the inverse of the distribution function. More...

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

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 \(\mu\) = `mu`. More...

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

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

static double	getStandardDeviation (double mu, double sigma)
	Computes the standard deviation of the half-normal distribution with parameters \(\mu\) and \(\sigma\). More...

Protected Attributes
double	mu

double	sigma

double	C1

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 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}

Member Function Documentation

◆ barF() [1/2]

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.

◆ barF() [2/2]

static double 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

◆ cdf() [1/2]

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.

◆ cdf() [2/2]

static double 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

◆ density()

static double 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

◆ getMean()

static double 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

◆ getMLE() [1/2]

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

◆ getMLE() [2/2]

static double [] 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 [ \(\sigma\)]

◆ getMu()

double getMu ( )

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

Returns: returns the parameter mu

◆ getParams()

double [] 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 Distribution.

◆ getSigma()

double getSigma ( )

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

Returns: returns the parameter sigma

◆ getStandardDeviation()

static double 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

◆ getVariance()

static double 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

◆ inverseF() [1/2]

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.

◆ inverseF() [2/2]

static double 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

◆ setParams()

void setParams	(	double	mu,
		double	sigma
	)

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

Parameters

mu	the parameter mu
sigma	the parameter sigma

◆ toString()

String toString ( )

Returns a String containing information about the current distribution.

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

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

HalfNormalDist.java

Public Member Functions

Static Public Member Functions

Protected Attributes

Additional Inherited Members

Detailed Description

Member Function Documentation

◆ barF() [1/2]

◆ barF() [2/2]

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ density()

◆ getMean()

◆ getMLE() [1/2]

◆ getMLE() [2/2]

◆ getMu()

◆ getParams()

◆ getSigma()

◆ getStandardDeviation()

◆ getVariance()

◆ inverseF() [1/2]

◆ inverseF() [2/2]

◆ setParams()

◆ toString()