umontreal.iro.lecuyer.probdist
Class ChiDist

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

All Implemented Interfaces:

Distribution

public class ChiDist
extends ContinuousDistribution

Extends the class ContinuousDistribution for the chi distribution with shape parameter v > 0, where the number of degrees of freedom v is a positive integer. The density function is given by

f (x) = e^-x2/2x^v-1/(2^(v/2)-1Γ(v/2)) for x > 0,

where Γ(x) is the gamma function defined in GammaDist. The distribution function is

F(x) = 1/Γ(v/2)∫₀^x2/2t^v/2-1e^-t dt.

It is equivalent to the gamma distribution function with parameters α = v/2 and λ = 1, evaluated at x²/2.

Field Summary

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

decPrec

Constructor Summary

ChiDist(int nu)
Constructs a ChiDist object.

Method Summary

double	barF(double x) Returns the complementary distribution function.
static double	barF(int nu, double x) Computes the complementary distribution.
double	cdf(double x) Returns the distribution function F(x).
static double	cdf(int nu, double x) Computes the distribution function by using the gamma distribution function.
double	density(double x) Returns f (x), the density evaluated at x.
static double	density(int nu, double x) Computes the density function.
static ChiDist	getInstanceFromMLE(double[] x, int n) Creates a new instance of a chi distribution with parameter ν estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.
double	getMean() Returns the mean.
static double	getMean(int nu) Computes and returns the mean of the chi distribution with parameter ν.
static double[]	getMLE(double[] x, int n) Estimates the parameter ν of the chi distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1.
int	getNu() Returns the value of ν for this object.
double[]	getParams() Return a table containing parameters of the current distribution.
double	getStandardDeviation() Returns the standard deviation.
static double	getStandardDeviation(int nu) Computes and returns the standard deviation of the chi distribution with parameter ν.
double	getVariance() Returns the variance.
static double	getVariance(int nu) Computes and returns the variance of the chi distribution with parameter ν.
double	inverseF(double u) Returns the inverse distribution function x = F-1(u).
static double	inverseF(int nu, double u) Returns the inverse distribution function computed using the gamma inversion.
void	setNu(int nu) Sets the value of ν for this object.
String	toString()

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

ChiDist

public ChiDist(int nu)

Constructs a ChiDist object.

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(int nu,
                             double x)

Computes the density function.

cdf

public static double cdf(int nu,
                         double x)

Computes the distribution function by using the gamma distribution function.

barF

public static double barF(int nu,
                          double x)

Computes the complementary distribution.

inverseF

public static double inverseF(int nu,
                              double u)

Returns the inverse distribution function computed using the gamma inversion.

getMLE

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

Estimates the parameter ν 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(ν)]

getInstanceFromMLE

public static ChiDist getInstanceFromMLE(double[] x,
                                         int n)

Creates a new instance of a chi distribution with parameter ν 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

public static double getMean(int nu)

Computes and returns the mean of the chi distribution with parameter ν.

Returns:

the mean of the chi distribution E[X] = (2)^1/2Γ((ν +1)/2)/Γ(ν/2)

getVariance

public static double getVariance(int nu)

Computes and returns the variance of the chi distribution with parameter ν.

Returns:

the variance of the chi distribution Var[X] = 2[Γ(ν/2)Γ(1 + ν/2) - Γ²(1/2(ν +1))]/Γ(ν/2)

getStandardDeviation

public static double getStandardDeviation(int nu)

Computes and returns the standard deviation of the chi distribution with parameter ν.

Returns:

the standard deviation of the chi distribution

getNu

public int getNu()

Returns the value of ν for this object.

setNu

public void setNu(int nu)

Sets the value of ν for this object.

getParams

public double[] getParams()

Return a table containing parameters of the current distribution.

toString

public String toString()

Overrides:

toString in class Object