umontreal.iro.lecuyer.probdist

Class ChiSquareNoncentralDist

java.lang.Object

umontreal.iro.lecuyer.probdist.ContinuousDistribution

umontreal.iro.lecuyer.probdist.ChiSquareNoncentralDist

All Implemented Interfaces:

Distribution

public class ChiSquareNoncentralDist
extends ContinuousDistribution

Extends the class ContinuousDistribution for the noncentral chi-square distribution with ν degrees of freedom and noncentrality parameter λ, where ν > 0 and λ > 0. Its density is

f (x) = (e^-(x+λ)/2)/2(x/λ)^(ν-2)/4I_ν/2-1((λx)^1/2) for x > 0,

where I_ν(x) is the modified Bessel function of the first kind of order ν given by

I_ν(z) = ∑_j=0^∞[(z/2)^ν+2j]/[j! Γ(ν + j + 1)],

where Γ(x) is the gamma function. Notice that this distribution is more general than the chi-square distribution since its number of degrees of freedom can be any positive real number. For λ = 0 and ν a positive integer, we have the ordinary chi-square distribution.

The cumulative probability function can be written as

P[X <= x] = ∑_j=0^∞(e^-λ/2(λ/2)^j/j!)P[χ²_ν+2j <= x],

where χ²_ν+2j is the central chi-square distribution with ν + 2j degrees of freedom.

Field Summary

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

decPrec

Constructor Summary

Constructors

Constructor and Description
ChiSquareNoncentralDist(double nu, double lambda) Constructs a noncentral chi-square distribution with ν = nu degrees of freedom and noncentrality parameter λ = lambda.

Method Summary

Modifier and Type	Method and Description
double	barF(double x) Returns the complementary distribution function.
static double	barF(double nu, double lambda, double x) Computes the complementary noncentral chi-square distribution function with ν = nu degrees of freedom and parameter λ = lambda.
double	cdf(double x) Returns the distribution function F(x).
static double	cdf(double nu, double lambda, double x) Computes the noncentral chi-square distribution function with ν = nu degrees of freedom and parameter λ = lambda.
double	density(double x) Returns f (x), the density evaluated at x.
static double	density(double nu, double lambda, double x) Computes the density function for a noncentral chi-square distribution with ν = nu degrees of freedom and parameter λ = lambda.
double	getLambda() Returns the parameter λ of this object.
double	getMean() Returns the mean.
static double	getMean(double nu, double lambda) Computes and returns the mean E[X] = ν + λ of the noncentral chi-square distribution with parameters ν = nu and λ = lambda.
double	getNu() Returns the parameter ν of this object.
double[]	getParams() Returns a table containing the parameters of the current distribution.
double	getStandardDeviation() Returns the standard deviation.
static double	getStandardDeviation(double nu, double lambda) Computes and returns the standard deviation of the noncentral chi-square distribution with parameters ν = nu and λ = lambda.
double	getVariance() Returns the variance.
static double	getVariance(double nu, double lambda) Computes and returns the variance Var[X] = 2(ν +2λ) of the noncentral chi-square distribution with parameters ν = nu and λ = lambda.
double	inverseF(double u) Returns the inverse distribution function x = F-1(u).
static double	inverseF(double nu, double lambda, double u) Computes the inverse of the noncentral chi-square distribution with ν = nu degrees of freedom and parameter λ = lambda.
void	setParams(double nu, double lambda) Sets the parameters ν = nu and λ = lambda of this object.
java.lang.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

ChiSquareNoncentralDist

public ChiSquareNoncentralDist(double nu,
                               double lambda)

Constructs a noncentral chi-square distribution with ν = nu degrees of freedom and noncentrality parameter λ = lambda.

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(double nu,
                             double lambda,
                             double x)

Computes the density function for a noncentral chi-square distribution with ν = nu degrees of freedom and parameter λ = lambda.

cdf

public static double cdf(double nu,
                         double lambda,
                         double x)

Computes the noncentral chi-square distribution function with ν = nu degrees of freedom and parameter λ = lambda.

barF

public static double barF(double nu,
                          double lambda,
                          double x)

Computes the complementary noncentral chi-square distribution function with ν = nu degrees of freedom and parameter λ = lambda.

inverseF

public static double inverseF(double nu,
                              double lambda,
                              double u)

Computes the inverse of the noncentral chi-square distribution with ν = nu degrees of freedom and parameter λ = lambda.

getMean

public static double getMean(double nu,
                             double lambda)

Computes and returns the mean E[X] = ν + λ of the noncentral chi-square distribution with parameters ν = nu and λ = lambda.

Returns:

the mean of the Noncentral noncentral chi-square distribution

getVariance

public static double getVariance(double nu,
                                 double lambda)

Computes and returns the variance Var[X] = 2(ν +2λ) of the noncentral chi-square distribution with parameters ν = nu and λ = lambda.

Returns:

the variance of the noncentral chi-square distribution

getStandardDeviation

public static double getStandardDeviation(double nu,
                                          double lambda)

Computes and returns the standard deviation of the noncentral chi-square distribution with parameters ν = nu and λ = lambda.

Returns:

the standard deviation of the noncentral chi-square distribution

getNu

public double getNu()

Returns the parameter ν of this object.

getLambda

public double getLambda()

Returns the parameter λ of this object.

setParams

public void setParams(double nu,
                      double lambda)

Sets the parameters ν = nu and λ = lambda of this object.

getParams

public double[] getParams()

Returns a table containing the parameters of the current distribution.

toString

public java.lang.String toString()

Overrides:

toString in class java.lang.Object