ChiDist (Java Libraries for Stochastic Simulation)

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

Constructors
Constructor and Description

ChiDist(int nu)
Constructs a ChiDist object.

Constructors
Constructor and Description
`ChiDist(int nu)` Constructs a `ChiDist` object.

Method Summary

Methods
Modifier and Type	Method and Description
`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.
`static double[]`	`getMaximumLikelihoodEstimate(double[] x, int n)` Deprecated.
`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.
`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
  - 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)]
  - getMaximumLikelihoodEstimate
```
@Deprecated
public static double[] getMaximumLikelihoodEstimate(double[] x,
                                               int n)
```
    Deprecated.
    
    Same as getMLE.
  - 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 java.lang.String toString()
```
    Overrides:
    
    toString in class java.lang.Object

Class ChiDist

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

ChiDist

Method Detail

density

cdf

barF

inverseF

getMean

getVariance

getStandardDeviation

density

cdf

barF

inverseF

getMLE

getMaximumLikelihoodEstimate

getInstanceFromMLE

getMean

getVariance

getStandardDeviation

getNu

setNu

getParams

toString