SSJ
V. 2.6.

## 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 ChiDistextends 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-x2/2xv-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)∫0x2/2tv/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`

SSJ
V. 2.6.

To submit a bug or ask questions, send an e-mail to Pierre L'Ecuyer.