umontreal.iro.lecuyer.probdist
Class ChiSquareDist

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

All Implemented Interfaces:

Distribution

Direct Known Subclasses:

ChiSquareDistQuick

public class ChiSquareDist
extends ContinuousDistribution

Extends the class ContinuousDistribution for the chi-square distribution with n degrees of freedom, where n is a positive integer. Its density is

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

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

Field Summary

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

decPrec

Constructor Summary

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

Method Summary

double	barF(double x) Returns the complementary distribution function.
static double	barF(int n, int d, double x) Computes the complementary chi-square distribution function with n degrees of freedom, evaluated at x.
double	cdf(double x) Returns the distribution function F(x).
static double	cdf(int n, int d, double x) Computes the chi-square distribution function with n degrees of freedom, evaluated at x.
double	density(double x) Returns f (x), the density evaluated at x.
static double	density(int n, double x) Computes the density function for a chi-square distribution with n degrees of freedom.
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.
double	getMean() Returns the mean.
static double	getMean(int n) Computes and returns the mean E[X] = n of the chi-square distribution with parameter n.
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.
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.
int	getN() Returns the parameter n of this object.
double[]	getParams() Return a table containing the parameters of the current distribution.
double	getStandardDeviation() Returns the standard deviation.
static double	getStandardDeviation(int n) Returns the standard deviation of the chi-square distribution with parameter n.
double	getVariance() Returns the variance.
static double	getVariance(int n) Returns the variance Var[X] = 2n of the chi-square distribution with parameter n.
double	inverseF(double u) Returns the inverse distribution function x = F-1(u).
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.
void	setN(int n) Sets the parameter n of 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

ChiSquareDist

public ChiSquareDist(int n)

Constructs a chi-square distribution with n degrees of freedom.

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 n,
                             double x)

Computes the density function for a chi-square distribution with n degrees of freedom.

cdf

public static double cdf(int n,
                         int d,
                         double x)

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.

barF

public static double barF(int n,
                          int d,
                          double x)

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.

inverseF

public 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. 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.

getMLE

public 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. 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)]

getInstanceFromMLE

public 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.

Parameters:

x - the list of observations to use to evaluate parameters

m - the number of observations to use to evaluate parameters

getMean

public static double getMean(int n)

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

getMomentsEstimate

public 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.

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)]

getVariance

public static double getVariance(int n)

Returns the variance Var[X] = 2n of the chi-square distribution with parameter n.

Returns:

the variance of the chi-square distribution VarX] = 2n

getStandardDeviation

public static double getStandardDeviation(int n)

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

Returns:

the standard deviation of the chi-square distribution

getN

public int getN()

Returns the parameter n of this object.

setN

public void setN(int n)

Sets the parameter n of this object.

getParams

public double[] getParams()

Return a table containing the parameters of the current distribution.

toString

public String toString()

Overrides:

toString in class Object