Extends the class ContinuousDistribution for the chi distribution. More...

Inheritance diagram for umontreal.ssj.probdist.ChiDist:

Public Member Functions
	ChiDist (int nu)
	Constructs a ChiDist object.
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.
int	getNu ()
	Returns the value of \(\nu\) for this object.
void	setNu (int nu)
	Sets the value of \(\nu\) for this object.
double[]	getParams ()
	Return a table containing 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 (int nu, double x)
	Computes the density function.
static double	cdf (int nu, double x)
	Computes the distribution function by using the gamma distribution function.
static double	barF (int nu, double x)
	Computes the complementary distribution.
static double	inverseF (int nu, double u)
	Returns the inverse distribution function computed using the gamma inversion.
static double[]	getMLE (double[] x, int n)
	Estimates the parameter \(\nu\) of the chi distribution using the maximum likelihood method, from the \(n\) observations \(x[i]\),.
static ChiDist	getInstanceFromMLE (double[] x, int n)
	Creates a new instance of a chi distribution with parameter.
static double	getMean (int nu)
	Computes and returns the mean.
static double	getVariance (int nu)
	Computes and returns the variance.
static double	getStandardDeviation (int nu)
	Computes and returns the standard deviation of the chi distribution with parameter \(\nu\).

Detailed Description

Extends the class ContinuousDistribution for the chi distribution.

[95] (page 417) with shape parameter \(\nu > 0\), where the number of degrees of freedom \(\nu\) is a positive integer. The density function is given by

\[ f (x) = \frac{e^{-x^2 /2} x^{\nu-1}}{2^{(\nu/2)-1}\Gamma(\nu/2)}, \qquad\mbox{ for } x > 0, \tag{Fchi} \]

where \(\Gamma(x)\) is the gamma function defined in ( Gamma ). The distribution function is

\[ F (x) = \frac{1}{\Gamma(\nu/2)} \int_0^{x^2/2} t^{\nu/2-1}e^{-t} dt. \]

It is equivalent to the gamma distribution function with parameters \(\alpha=\nu/2\) and \(\lambda=1\), evaluated at \(x^2/2\).

               <div class="SSJ-bigskip"></div>

Definition at line 49 of file ChiDist.java.

Constructor & Destructor Documentation

◆ ChiDist()

umontreal.ssj.probdist.ChiDist.ChiDist ( int nu )

Constructs a ChiDist object.

Definition at line 72 of file ChiDist.java.

Member Function Documentation

◆ barF() [1/2]

double umontreal.ssj.probdist.ChiDist.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 86 of file ChiDist.java.

◆ barF() [2/2]

double umontreal.ssj.probdist.ChiDist.barF	(	int	nu,
		double	x )

static

Computes the complementary distribution.

Definition at line 129 of file ChiDist.java.

◆ cdf() [1/2]

double umontreal.ssj.probdist.ChiDist.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 82 of file ChiDist.java.

◆ cdf() [2/2]

double umontreal.ssj.probdist.ChiDist.cdf	(	int	nu,
		double	x )

static

Computes the distribution function by using the gamma distribution function.

Definition at line 120 of file ChiDist.java.

◆ density() [1/2]

double umontreal.ssj.probdist.ChiDist.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 76 of file ChiDist.java.

◆ density() [2/2]

double umontreal.ssj.probdist.ChiDist.density	(	int	nu,
		double	x )

static

Computes the density function.

Definition at line 109 of file ChiDist.java.

◆ getInstanceFromMLE()

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

static

Creates a new instance of a chi distribution with parameter.

\(\nu\) 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 194 of file ChiDist.java.

◆ getMean() [1/2]

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

Returns the mean.

Returns: the mean

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 94 of file ChiDist.java.

◆ getMean() [2/2]

double umontreal.ssj.probdist.ChiDist.getMean ( int nu )

static

Computes and returns the mean.

\[ E[X] = \frac{\sqrt{2} \Gamma( \frac{\nu+ 1}{2} )}{\Gamma(\frac{\nu}{2})} \]

of the chi distribution with parameter \(\nu\).

Returns: the mean of the chi distribution \(E[X] = \sqrt{2}\Gamma((\nu+ 1) / 2) / \Gamma(\nu/ 2)\)

Definition at line 207 of file ChiDist.java.

◆ getMLE()

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

static

Estimates the parameter \(\nu\) of the chi distribution using the maximum likelihood method, from the \(n\) observations \(x[i]\),.

\(i = 0, 1, …, n-1\). The estimate is returned in element 0 of the returned array.

Parameters

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

Returns: returns the parameter [ \(\hat{\nu}\)]

Definition at line 153 of file ChiDist.java.

◆ getNu()

int umontreal.ssj.probdist.ChiDist.getNu ( )

Returns the value of \(\nu\) for this object.

Definition at line 243 of file ChiDist.java.

◆ getParams()

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

Return a table containing parameters of the current distribution.

Implements umontreal.ssj.probdist.Distribution.

Definition at line 261 of file ChiDist.java.

◆ getStandardDeviation() [1/2]

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

Returns the standard deviation.

Returns: the standard deviation

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 102 of file ChiDist.java.

◆ getStandardDeviation() [2/2]

double umontreal.ssj.probdist.ChiDist.getStandardDeviation ( int nu )

static

Computes and returns the standard deviation of the chi distribution with parameter \(\nu\).

Returns: the standard deviation of the chi distribution

Definition at line 236 of file ChiDist.java.

◆ getVariance() [1/2]

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

Returns the variance.

Returns: the variance

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 98 of file ChiDist.java.

◆ getVariance() [2/2]

double umontreal.ssj.probdist.ChiDist.getVariance ( int nu )

static

Computes and returns the variance.

\[ \mbox{Var}[X] = \frac{2 \Gamma(\frac{\nu}{2}) \Gamma(1 + \frac{\nu}{2}) - \Gamma^2(\frac{\nu+ 1}{2})}{\Gamma(\frac{\nu}{2})} \]

of the chi distribution with parameter \(\nu\).

Returns: the variance of the chi distribution \(\mbox{Var}[X] = 2 [ \Gamma(\nu/ 2) \Gamma(1 + \nu/ 2) - \Gamma^2(1/2 (\nu+ 1)) ] / \Gamma(\nu/ 2)\)

Definition at line 223 of file ChiDist.java.

◆ inverseF() [1/2]

double umontreal.ssj.probdist.ChiDist.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 90 of file ChiDist.java.

◆ inverseF() [2/2]

double umontreal.ssj.probdist.ChiDist.inverseF	(	int	nu,
		double	u )

static

Returns the inverse distribution function computed using the gamma inversion.

Definition at line 138 of file ChiDist.java.

◆ setNu()

void umontreal.ssj.probdist.ChiDist.setNu ( int nu )

Sets the value of \(\nu\) for this object.

Definition at line 250 of file ChiDist.java.

◆ toString()

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

Returns a String containing information about the current distribution.

Definition at line 269 of file ChiDist.java.

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

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

SSJ API Documentation Stochastic Simulation in Java

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ ChiDist()

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()

◆ getNu()

◆ getParams()

◆ getStandardDeviation() [1/2]

◆ getStandardDeviation() [2/2]

◆ getVariance() [1/2]

◆ getVariance() [2/2]

◆ inverseF() [1/2]

◆ inverseF() [2/2]

◆ setNu()

◆ toString()