Extends the class ContinuousDistribution for the chi distribution [99] (page 417) with shape parameter \(\nu > 0\), where the number of degrees of freedom \(\nu\) is a positive integer. More...

Inheritance diagram for ChiDist:

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Collaboration diagram for ChiDist:

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Public Member Functions
	ChiDist (int nu)
	Constructs a `ChiDist` object.

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.

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

static ChiDist	getInstanceFromMLE (double[] x, int n)
	Creates a new instance of a chi distribution with parameter \(\nu\) estimated using the maximum likelihood method based on the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\). More...

static double	getMean (int nu)
	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 \(\nu\). More...

static double	getVariance (int nu)
	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 \(\nu\). More...

static double	getStandardDeviation (int nu)
	Computes and returns the standard deviation of the chi distribution with parameter \(\nu\). More...

Additional Inherited Members
Public Attributes inherited from ContinuousDistribution
int	decPrec = 15

Protected Attributes inherited from ContinuousDistribution
double	supportA = Double.NEGATIVE_INFINITY

double	supportB = Double.POSITIVE_INFINITY

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 chi distribution [99] (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 \(x^2/2\).

Member Function Documentation

◆ barF()

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.

◆ cdf()

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.

◆ getInstanceFromMLE()

static 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

◆ getMean()

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

◆ getMLE()

static double [] 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 [ \(\hat{\nu}\)]

◆ getStandardDeviation()

static double 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

◆ getVariance()

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

◆ inverseF()

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.

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

ChiDist.java

Public Member Functions

Static Public Member Functions

Additional Inherited Members

Detailed Description

Member Function Documentation

◆ barF()

◆ cdf()

◆ getInstanceFromMLE()

◆ getMean()

◆ getMLE()

◆ getStandardDeviation()

◆ getVariance()

◆ inverseF()