umontreal.iro.lecuyer.probdistmulti

Class BiStudentDist

java.lang.Object

umontreal.iro.lecuyer.probdistmulti.ContinuousDistributionMulti

umontreal.iro.lecuyer.probdistmulti.ContinuousDistribution2Dim

umontreal.iro.lecuyer.probdistmulti.BiStudentDist

public class BiStudentDist
extends ContinuousDistribution2Dim

Extends the class ContinuousDistribution2Dim for the standard bivariate Student's t distribution. The correlation between X and Y is r and the number of degrees of freedom is ν. Its probability density is

f (x, y) = (1 + (x² -2ρxy + y²)/(ν(1 - ρ²)))^-(ν+2)/2/(2π(1-r^2)^1/2),

and the corresponding distribution function (the cdf) is

T_ν(x, y, r) = ∫_-∞^xdx∫_-∞^ydy f (x, y)/(2π(1 - r^2)^1/2).

We also define the upper distribution function called barF as

bar(T)_ν(x, y, r) = ∫^∞_xdx∫^∞_ydy f (x, y)/(2π(1 - r^2)^1/2).

Field Summary

Fields inherited from class umontreal.iro.lecuyer.probdistmulti.ContinuousDistribution2Dim

decPrec

Constructor Summary

Constructors

Constructor and Description
BiStudentDist(int nu, double rho) Constructs a BiStudentDist object with correlation ρ = rho and ν = nu degrees of freedom.

Method Summary

Modifier and Type	Method and Description
double	barF(double x, double y) .
static double	barF(int nu, double x, double y, double rho) Computes the standard upper bivariate Student's t distribution.
double	cdf(double x, double y) .
static double	cdf(int nu, double x, double y, double rho) Computes the standard bivariate Student's t distribution using the method described in.
double	density(double x, double y) Returns f (x, y), the density of (X, Y) evaluated at (x, y).
static double	density(int nu, double x, double y, double rho) Computes the standard bivariate Student's t density function with correlation ρ = rho and ν = nu degrees of freedom.
double[][]	getCorrelation() Returns the correlation matrix of the distribution, defined as ρij = σij/(σ_iiσ_jj)1/2.
static double[][]	getCorrelation(int nu, double rho) Returns the correlation matrix of the bivariate Student's t distribution.
double[][]	getCovariance() Returns the variance-covariance matrix of the distribution, defined as σij = E[(Xi - μi)(Xj - μj)].
static double[][]	getCovariance(int nu, double rho) Returns the covariance matrix of the bivariate Student's t distribution.
double[]	getMean() Returns the mean vector of the distribution, defined as μi = E[Xi].
static double[]	getMean(int nu, double rho) Returns the mean vector E[X] = (0, 0) of the bivariate Student's t distribution.

Methods inherited from class umontreal.iro.lecuyer.probdistmulti.ContinuousDistribution2Dim

cdf, density

Methods inherited from class umontreal.iro.lecuyer.probdistmulti.ContinuousDistributionMulti

getDimension

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

BiStudentDist

public BiStudentDist(int nu,
                     double rho)

Constructs a BiStudentDist object with correlation ρ = rho and ν = nu degrees of freedom.

Method Detail

density

public double density(double x,
                      double y)

Description copied from class: ContinuousDistribution2Dim

Returns f (x, y), the density of (X, Y) evaluated at (x, y).

Specified by:

density in class ContinuousDistribution2Dim

Parameters:

x - value x at which the density is evaluated

y - value y at which the density is evaluated

Returns:

density function evaluated at (x, y)

cdf

public double cdf(double x,
                  double y)

Description copied from class: ContinuousDistribution2Dim

. Computes the distribution function F(x, y):

F(x, y) = P[X <= x, Y <= y] = ∫_-∞^xds∫_-∞^ydt f (s, t).

Specified by:

cdf in class ContinuousDistribution2Dim

Parameters:

x - value x at which the distribution function is evaluated

y - value y at which the distribution function is evaluated

Returns:

distribution function evaluated at (x, y)

barF

public double barF(double x,
                   double y)

Description copied from class: ContinuousDistribution2Dim

. Computes the upper cumulative distribution function bar(F)(x, y):

bar(F)(x, y) = P[X >= x, Y >= y] = ∫^∞_xds∫^∞_ydt f (s, t).

Overrides:

barF in class ContinuousDistribution2Dim

Parameters:

x - value x at which the upper distribution is evaluated

y - value y at which the upper distribution is evaluated

Returns:

upper distribution function evaluated at (x, y)

density

public static double density(int nu,
                             double x,
                             double y,
                             double rho)

Computes the standard bivariate Student's t density function with correlation ρ = rho and ν = nu degrees of freedom.

cdf

public static double cdf(int nu,
                         double x,
                         double y,
                         double rho)

Computes the standard bivariate Student's t distribution using the method described in. The code for the cdf was translated directly from the Matlab code written by Alan Genz and available from his web page at http://www.math.wsu.edu/faculty/genz/homepage (the code is copyrighted by Alan Genz and is included in this package with the kind permission of the author). The correlation is ρ = rho and the number of degrees of freedom is ν = nu.

barF

public static double barF(int nu,
                          double x,
                          double y,
                          double rho)

Computes the standard upper bivariate Student's t distribution.

getMean

public double[] getMean()

Description copied from class: ContinuousDistributionMulti

Returns the mean vector of the distribution, defined as μ_i = E[X_i].

Specified by:

getMean in class ContinuousDistributionMulti

getMean

public static double[] getMean(int nu,
                               double rho)

Returns the mean vector E[X] = (0, 0) of the bivariate Student's t distribution.

getCovariance

public double[][] getCovariance()

Description copied from class: ContinuousDistributionMulti

Returns the variance-covariance matrix of the distribution, defined as
σ_ij = E[(X_i - μ_i)(X_j - μ_j)].

Specified by:

getCovariance in class ContinuousDistributionMulti

getCovariance

public static double[][] getCovariance(int nu,
                                       double rho)

Returns the covariance matrix of the bivariate Student's t distribution.

getCorrelation

public double[][] getCorrelation()

Description copied from class: ContinuousDistributionMulti

Returns the correlation matrix of the distribution, defined as ρ_ij = σ_ij/(σ_iiσ_jj)^1/2.

Specified by:

getCorrelation in class ContinuousDistributionMulti

getCorrelation

public static double[][] getCorrelation(int nu,
                                        double rho)

Returns the correlation matrix of the bivariate Student's t distribution.