umontreal.iro.lecuyer.probdist
Class StudentDist

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

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) Computes and 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 of X evaluated at x.
static double	density(int n, double x) Computes the density function for a Student-t distribution with n degrees of freedom.
int	getN() Returns the parameter n associated with this object.
double	inverseF(double u) Computes and returns the inverse distribution function F-1(u), defined in.
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.

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

barF, inverseBisection, inverseBrent

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, toString, 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 of X 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

Computes and 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 interface: Distribution

Computes and returns the inverse distribution function F^-1(u), defined in.

Specified by:

inverseF in interface Distribution

Overrides:

inverseF in class ContinuousDistribution

Parameters:

u - value in the interval (0, 1) for which the inverse distribution function is evaluated

Returns:

the inverse distribution function evaluated at u

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 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 .

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.