ChiSquareNoncentralDist (Java Libraries for Stochastic Simulation)

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 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

Methods
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

Class ChiSquareNoncentralDist

Field Summary

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

Constructor Summary

Method Summary

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

Methods inherited from class java.lang.Object

Constructor Detail

ChiSquareNoncentralDist

Method Detail

density

cdf

barF

inverseF

getMean

getVariance

getStandardDeviation

density

cdf

barF

inverseF

getMean

getVariance

getStandardDeviation

getNu

getLambda

setParams

getParams

toString