Extends the class ContinuousDistribution for the chi-square distribution with \(n\) degrees of freedom, where \(n\) is a positive integer [99] (page 416). More...

Inheritance diagram for ChiSquareDist:

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Collaboration diagram for ChiSquareDist:

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Public Member Functions
	ChiSquareDist (int n)
	Constructs a chi-square distribution with `n` degrees of freedom.

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	getN ()
	Returns the parameter \(n\) of this object.

void	setN (int n)
	Sets the parameter \(n\) of this object.

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 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 n, double x)
	Computes the density function ( Fchi2 ) for a chi-square distribution with \(n\) degrees of freedom.

static double	cdf (int n, int d, double x)
	Computes the chi-square distribution function with \(n\) degrees of freedom, evaluated at \(x\). More...

static double	barF (int n, int d, double x)
	Computes the complementary chi-square distribution function with \(n\) degrees of freedom, evaluated at \(x\). More...

static double	inverseF (int n, double u)
	Computes an approximation of \(F^{-1}(u)\), where \(F\) is the chi-square distribution with \(n\) degrees of freedom. More...

static double []	getMLE (double[] x, int m)
	Estimates the parameter \(n\) of the chi-square distribution using the maximum likelihood method, from the \(m\) observations \(x[i]\), \(i = 0, 1, …, m-1\). More...

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

static double	getMean (int n)
	Computes and returns the mean \(E[X] = n\) of the chi-square distribution with parameter \(n\). More...

static double []	getMomentsEstimate (double[] x, int m)
	Estimates and returns the parameter [ \(\hat{n}\)] of the chi-square distribution using the moments method based on the \(m\) observations in table \(x[i]\), \(i = 0, 1, …, m-1\). More...

static double	getVariance (int n)
	Returns the variance \(\mbox{Var}[X] = 2n\) of the chi-square distribution with parameter \(n\). More...

static double	getStandardDeviation (int n)
	Returns the standard deviation of the chi-square distribution with parameter \(n\). More...

Protected Attributes
int	n

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 chi-square distribution with \(n\) degrees of freedom, where \(n\) is a positive integer [99] (page 416).

Its density is

\[ f(x) = \frac{x^{(n/2)-1}e^{-x/2}}{2^{n/2}\Gamma(n/2)},\qquad\mbox{for } x > 0 \tag{Fchi2} \]

where \(\Gamma(x)\) is the gamma function defined in ( Gamma ). The chi-square distribution is a special case of the gamma distribution with shape parameter \(n/2\) and scale parameter \(1/2\). Therefore, one can use the methods of GammaDist for this distribution.

The non-static versions of the methods cdf, barF, and inverseF call the static version of the same name.

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	(	int	n,
		int	d,
		double	x
	)

static

Computes the complementary chi-square distribution function with \(n\) degrees of freedom, evaluated at \(x\).

The method tries to return \(d\) decimals digits of precision, but there is no guarantee.

◆ 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	(	int	n,
		int	d,
		double	x
	)

static

Computes the chi-square distribution function with \(n\) degrees of freedom, evaluated at \(x\).

The method tries to return \(d\) decimals digits of precision, but there is no guarantee.

◆ getInstanceFromMLE()

static ChiSquareDist getInstanceFromMLE	(	double []	x,
		int	m
	)

static

Creates a new instance of a chi-square distribution with parameter \(n\) estimated using the maximum likelihood method based on the \(m\) observations \(x[i]\), \(i = 0, 1, …, m-1\).

Parameters

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

◆ getMean()

static double getMean ( int n )

static

Computes and returns the mean \(E[X] = n\) of the chi-square distribution with parameter \(n\).

Returns: the mean of the Chi-square distribution \(E[X] = n\)

◆ getMLE()

static double [] getMLE	(	double []	x,
		int	m
	)

static

Estimates the parameter \(n\) of the chi-square distribution using the maximum likelihood method, from the \(m\) observations \(x[i]\), \(i = 0, 1, …, m-1\).

The estimate is returned in element 0 of the returned array.

Parameters

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

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

◆ getMomentsEstimate()

static double [] getMomentsEstimate	(	double []	x,
		int	m
	)

static

Estimates and returns the parameter [ \(\hat{n}\)] of the chi-square distribution using the moments method based on the \(m\) observations in table \(x[i]\), \(i = 0, 1, …, m-1\).

Parameters

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

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

◆ getStandardDeviation()

static double getStandardDeviation ( int n )

static

Returns the standard deviation of the chi-square distribution with parameter \(n\).

Returns: the standard deviation of the chi-square distribution

◆ getVariance()

static double getVariance ( int n )

static

Returns the variance \(\mbox{Var}[X] = 2n\) of the chi-square distribution with parameter \(n\).

Returns: the variance of the chi-square distribution \(\mbox{Var}X] = 2n\)

◆ 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	(	int	n,
		double	u
	)

static

Computes an approximation of \(F^{-1}(u)\), where \(F\) is the chi-square distribution with \(n\) degrees of freedom.

Uses the approximation given in [17] and in Figure L.23 of [25] . It gives at least 6 decimal digits of precision, except far in the tails (that is, for \(u< 10^{-5}\) or \(u > 1 - 10^{-5}\)) where the function calls the method GammaDist.inverseF (n/2, 7, u) and multiplies the result by 2.0. To get better precision, one may call GammaDist.inverseF, but this method is slower than the current method, especially for large \(n\). For instance, for \(n = \) 16, 1024, and 65536, the GammaDist.inverseF method is 2, 5, and 8 times slower, respectively, than the current method.

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

ChiSquareDist.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]

◆ getInstanceFromMLE()

◆ getMean()

◆ getMLE()

◆ getMomentsEstimate()

◆ getStandardDeviation()

◆ getVariance()

◆ inverseF() [1/2]

◆ inverseF() [2/2]