StudentDist (Java Libraries for Stochastic Simulation)

java.lang.Object
- umontreal.iro.lecuyer.probdist.ContinuousDistribution
- - umontreal.iro.lecuyer.probdist.StudentDist

All Implemented Interfaces:

Distribution
```
public class StudentDist
extends ContinuousDistribution
```
Extends the class ContinuousDistribution for the Student-t distribution with n degrees of freedom, where n is a positive integer. Its density is

f (x) = [Γ((n + 1)/2)/(Γ(n/2)(πn)^1/2)][1 + x²/n]^-(n+1)/2 for - ∞ < x < ∞,
where Γ(x) is the gamma function defined in GammaDist.
The non-static methods cdf and barF use the same algorithm as in cdf.

Field Summary
- Fields inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
  decPrec

Constructor Summary

Constructors
Constructor and Description

StudentDist(int n)
Constructs a StudentDist object with n degrees of freedom.

Constructors
Constructor and Description
`StudentDist(int n)` Constructs a `StudentDist` object with `n` degrees of freedom.

Method Summary

Methods
Modifier and Type	Method and Description
`static double`	`barF(int n, double x)` Computes the complementary distribution function bar(F)(x).
`double`	`cdf(double x)` Returns the distribution function F(x).
`static double`	`cdf(int n, double x)` Returns an approximation for the Student-t distribution function with n degrees of freedom.
`static double`	`cdf2(int n, int d, double x)` Returns an approximation of the Student-t distribution function with n degrees of freedom.
`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 Student-t distribution with n degrees of freedom.
`static StudentDist`	`getInstanceFromMLE(double[] x, int m)` Creates a new instance of a Student-t distribution with parameter n estimated using the maximum likelihood method based on the m observations x[i], i = 0, 1,…, m - 1.
`static double[]`	`getMaximumLikelihoodEstimate(double[] x, int m)` Deprecated.
`double`	`getMean()` Returns the mean.
`static double`	`getMean(int n)` Returns the mean E[X] = 0 of the Student-t distribution with parameter n.
`static double[]`	`getMLE(double[] x, int m)` Estimates the parameter n of the Student-t distribution using the maximum likelihood method, from the m observations x[i], i = 0, 1,…, m - 1.
`int`	`getN()` Returns the parameter n associated with this object.
`double[]`	`getParams()` Return a table containing the parameter of the current distribution.
`double`	`getStandardDeviation()` Returns the standard deviation.
`static double`	`getStandardDeviation(int n)` Computes and returns the standard deviation of the Student-t distribution with parameter n.
`double`	`getVariance()` Returns the variance.
`static double`	`getVariance(int n)` Computes and returns the variance Var[X] = n/(n - 2) of the Student-t distribution with parameter n.
`double`	`inverseF(double u)` Returns the inverse distribution function x = F^-1(u).
`static double`	`inverseF(int n, double u)` Returns an approximation of F^-1(u), where F is the Student-t distribution function with n degrees of freedom.
`void`	`setN(int n)` Sets the parameter n associated with this object.
`java.lang.String`	`toString()`

Methods inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
barF, getXinf, getXsup, inverseBisection, inverseBrent, setXinf, setXsup

Methods inherited from class java.lang.Object
equals, getClass, hashCode, notify, notifyAll, wait, wait, wait

- Constructor Detail
  - StudentDist
```
public StudentDist(int n)
```
    Constructs a StudentDist object 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
  - 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 Student-t distribution with n degrees of freedom.
  - cdf
```
public static double cdf(int n,
         double x)
```
    Returns an approximation for the Student-t distribution function with n degrees of freedom. Gives at least 12 decimals of precision for n <= 10³, and at least 10 decimals for 10³ < n <= 10⁵.
  - cdf2
```
public static double cdf2(int n,
          int d,
          double x)
```
    Returns an approximation of the Student-t distribution function with n degrees of freedom. Uses the relationship
    
    2F(x) = I_{n/2, 1/2}(n/(n + x²)) for x < 0,
    
    2F(x) = I_{1/2, n/2}(x²/(n + x²)) for x >= 0,
    
    where I_{α, β} is the beta distribution function with parameters α and β (also called the incomplete beta ratio) defined in BetaDist, which is approximated by calling BetaDist.cdf. The function tries to return d decimals digits of precision (but there is no guarantee). This method is much slower (twenty to forty times, depending on parameters) than cdf, but could be used if precision is important.
  - barF
```
public static double barF(int n,
          double x)
```
    Computes the complementary distribution function bar(F)(x).
  - inverseF
```
public static double inverseF(int n,
              double u)
```
    Returns an approximation of F^-1(u), where F is the Student-t distribution function with n degrees of freedom. Uses an approximation giving at least 5 decimal digits of precision when n >= 8 or n <= 2, and 3 decimal digits of precision when 3 <= n <= 7 .
  - getMLE
```
public static double[] getMLE(double[] x,
              int m)
```
    Estimates the parameter n of the Student-t distribution using the maximum likelihood method, from the m observations x[i], i = 0, 1,…, m - 1. The estimate is returned in a one-element 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)]
  - getMaximumLikelihoodEstimate
```
@Deprecated
public static double[] getMaximumLikelihoodEstimate(double[] x,
                                               int m)
```
    Deprecated.
    
    Same as getMLE.
  - getInstanceFromMLE
```
public static StudentDist getInstanceFromMLE(double[] x,
                             int m)
```
    Creates a new instance of a Student-t 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)
```
    Returns the mean E[X] = 0 of the Student-t distribution with parameter n.
    
    Returns:
    the mean of the Student-t distribution E[X] = 0
  - getVariance
```
public static double getVariance(int n)
```
    Computes and returns the variance Var[X] = n/(n - 2) of the Student-t distribution with parameter n.
    
    Returns:
    the variance of the Student-t distribution Var[X] = n/(n - 2)
  - getStandardDeviation
```
public static double getStandardDeviation(int n)
```
    Computes and returns the standard deviation of the Student-t distribution with parameter n.
    
    Returns:
    the standard deviation of the Student-t distribution
  - getN
```
public int getN()
```
    Returns the parameter n associated with this object.
  - setN
```
public void setN(int n)
```
    Sets the parameter n associated with this object.
  - getParams
```
public double[] getParams()
```
    Return a table containing the parameter of the current distribution.
  - toString
```
public java.lang.String toString()
```
    Overrides:
    
    toString in class java.lang.Object

2F(x) =	I_{n/2, 1/2}(n/(n + x²))	for x < 0,
2F(x) =	I_{1/2, n/2}(x²/(n + x²))	for x >= 0,

Class StudentDist

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

StudentDist

Method Detail

density

cdf

inverseF

getMean

getVariance

getStandardDeviation

density

cdf

cdf2

barF

inverseF

getMLE

getMaximumLikelihoodEstimate

getInstanceFromMLE

getMean

getVariance

getStandardDeviation

getN

setN

getParams

toString