SSJ
V. 2.2.

umontreal.iro.lecuyer.probdist
Class StudentDist

java.lang.Object
  extended by umontreal.iro.lecuyer.probdist.ContinuousDistribution
      extended by 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
StudentDist(int n)
          Constructs a StudentDist object with n degrees of freedom.
 
Method Summary
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.
 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) = In/2, 1/2(n/(n + x2))          for x < 0,
2F(x) = I1/2, n/2(x2/(n + x2))          for x >= 0,

where Iα, 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 String toString()
Overrides:
toString in class Object

SSJ
V. 2.2.

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