umontreal.iro.lecuyer.probdist
Class HalfNormalDist

java.lang.Object
  umontreal.iro.lecuyer.probdist.ContinuousDistribution
      umontreal.iro.lecuyer.probdist.HalfNormalDist

All Implemented Interfaces:

Distribution

public class HalfNormalDist
extends ContinuousDistribution

Extends the class ContinuousDistribution for the half-normal distribution with parameters μ and σ > 0. Its density is

f (x) = ((2/π)^1/2/σ)e^{-(x-μ)2/(2σ2)}, for x > = μ,

f (x) = 0 for x < μ,

Field Summary

Fields inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution

decPrec

Constructor Summary

HalfNormalDist(double mu, double sigma)
Constructs a HalfNormalDist object with parameters μ = mu and σ = sigma.

Method Summary

double	barF(double x) Returns the complementary distribution function.
static double	barF(double mu, double sigma, double x) Computes the complementary distribution function.
double	cdf(double x) Returns the distribution function F(x).
static double	cdf(double mu, double sigma, double x) Computes the distribution function.
double	density(double x) Returns f (x), the density evaluated at x.
static double	density(double mu, double sigma, double x) Computes the density function of the half-normal distribution.
double	getMean() Returns the mean.
static double	getMean(double mu, double sigma) Computes and returns the mean E[X] = μ + σ(2 / π)1/2.
static double[]	getMLE(double[] x, int n) Estimates the parameters μ and σ 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 σ of the half-normal distribution using the maximum likelihood method from the n observations x[i], i = 0, 1,…, n - 1 and the parameter μ = mu.
double	getMu() Returns the parameter μ of this object.
double[]	getParams() Return a table containing the parameters of the current distribution.
double	getSigma() Returns the parameter σ of this object.
double	getStandardDeviation() Returns the standard deviation.
static double	getStandardDeviation(double mu, double sigma) Computes the standard deviation of the half-normal distribution with parameters μ and σ.
double	getVariance() Returns the variance.
static double	getVariance(double mu, double sigma) Computes and returns the variance Var[X] = (1 - 2/π)σ2.
double	inverseF(double u) Returns the inverse distribution function x = F-1(u).
static double	inverseF(double mu, double sigma, double u) Computes the inverse of the distribution function.
void	setParams(double mu, double sigma) Sets the parameters μ and σ.
String	toString() Returns a String containing information about the current distribution.

Methods inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution

getXinf, getXsup, inverseBisection, inverseBrent, setXinf, setXsup

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, wait, wait, wait

Constructor Detail

HalfNormalDist

public HalfNormalDist(double mu,
                      double sigma)

Constructs a HalfNormalDist object with parameters μ = mu and σ = sigma.

Method Detail

density

public double density(double x)

Description copied from class: ContinuousDistribution

Returns f (x), the density evaluated at x.

Specified by:

density in class ContinuousDistribution

Parameters:

x - value at which the density is evaluated

Returns:

density function evaluated at x

cdf

public double cdf(double x)

Description copied from interface: Distribution

Returns the distribution function F(x).

Parameters:

x - value at which the distribution function is evaluated

Returns:

distribution function evaluated at x

barF

public double barF(double x)

Description copied from class: ContinuousDistribution

Returns the complementary distribution function. The default implementation computes bar(F)(x) = 1 - F(x).

Specified by:

barF in interface Distribution

Overrides:

barF in class ContinuousDistribution

Parameters:

x - value at which the complementary distribution function is evaluated

Returns:

complementary distribution function evaluated at x

inverseF

public double inverseF(double u)

Description copied from class: ContinuousDistribution

Returns the inverse distribution function x = F^-1(u). Restrictions: u∈[0, 1].

Specified by:

inverseF in interface Distribution

Overrides:

inverseF in class ContinuousDistribution

Parameters:

u - value at which the inverse distribution function is evaluated

Returns:

the inverse distribution function evaluated at u

getMean

public double getMean()

Description copied from class: ContinuousDistribution

Returns the mean.

Specified by:

getMean in interface Distribution

Overrides:

getMean in class ContinuousDistribution

Returns:

the mean

getVariance

public double getVariance()

Description copied from class: ContinuousDistribution

Returns the variance.

Specified by:

getVariance in interface Distribution

Overrides:

getVariance in class ContinuousDistribution

Returns:

the variance

getStandardDeviation

public double getStandardDeviation()

Description copied from class: ContinuousDistribution

Returns the standard deviation.

Specified by:

getStandardDeviation in interface Distribution

Overrides:

getStandardDeviation in class ContinuousDistribution

Returns:

the standard deviation

density

public static double density(double mu,
                             double sigma,
                             double x)

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

cdf

public static double cdf(double mu,
                         double sigma,
                         double x)

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

barF

public static double barF(double mu,
                          double sigma,
                          double x)

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

inverseF

public static double inverseF(double mu,
                              double sigma,
                              double u)

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

getMLE

public static double[] getMLE(double[] x,
                              int n)

Estimates the parameters μ and σ 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: [μ, σ].

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 [μ, σ]

getMLE

public static double[] getMLE(double[] x,
                              int n,
                              double mu)

Estimates the parameter σ of the half-normal distribution using the maximum likelihood method from the n observations x[i], i = 0, 1,…, n - 1 and the parameter μ = mu. The estimate is returned in a one-element array: [σ].

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 [σ]

getMean

public static double getMean(double mu,
                             double sigma)

Computes and returns the mean E[X] = μ + σ(2 / π)^1/2.

Parameters:

mu - the parameter mu

sigma - the parameter sigma

Returns:

returns the mean

getVariance

public static double getVariance(double mu,
                                 double sigma)

Computes and returns the variance Var[X] = (1 - 2/π)σ².

Parameters:

mu - the parameter mu

sigma - the parameter sigma

Returns:

returns the variance

getStandardDeviation

public static double getStandardDeviation(double mu,
                                          double sigma)

Computes the standard deviation of the half-normal distribution with parameters μ and σ.

Parameters:

mu - the parameter mu

sigma - the parameter sigma

Returns:

returns the standard deviation

getMu

public double getMu()

Returns the parameter μ of this object.

Returns:

returns the parameter mu

getSigma

public double getSigma()

Returns the parameter σ of this object.

Returns:

returns the parameter sigma

setParams

public void setParams(double mu,
                      double sigma)

Sets the parameters μ and σ.

Parameters:

mu - the parameter mu

sigma - the parameter sigma

getParams

public double[] getParams()

Return a table containing the parameters of the current distribution. This table is put in regular order: [μ, σ].

Returns:

returns the parameters [μ, σ]

toString

public String toString()

Returns a String containing information about the current distribution.

Overrides:

toString in class Object

Returns:

returns a String containing information about the current distribution.