Extends the class ContinuousDistribution2Dim for the standard bivariate Student’s \(t\) distribution [94] (page 132). More...
Public Member Functions | |
| BiStudentDist (int nu, double rho) | |
| Constructs a BiStudentDist object with correlation \(\rho=
\) rho and \(\nu\) = nu degrees of freedom. | |
| double | density (double x, double y) |
| Returns \(f(x, y)\), the density of \((X, Y)\) evaluated at \((x, y)\). | |
| double | cdf (double x, double y) |
| Computes the distribution function \(F(x, y)\): | |
| double | barF (double x, double y) |
| Computes the upper cumulative distribution function. | |
| double[] | getMean () |
| Returns the mean vector of the distribution, defined as \(\mu_i = E[X_i]\). | |
| double[][] | getCovariance () |
| Returns the variance-covariance matrix of the distribution, defined as \(\sigma_{ij} = E[(X_i - \mu_i)(X_j - \mu_j)]\). | |
| double[][] | getCorrelation () |
| Returns the correlation matrix of the distribution, defined as. | |
| Public Member Functions inherited from umontreal.ssj.probdistmulti.ContinuousDistribution2Dim | |
| double | density (double[] x) |
| Simply calls density (x[0], x[1]). | |
| double | cdf (double a1, double a2, double b1, double b2) |
| Computes the cumulative probability in the square region. | |
| Public Member Functions inherited from umontreal.ssj.probdistmulti.ContinuousDistributionMulti | |
| int | getDimension () |
| Returns the dimension \(d\) of the distribution. | |
Static Public Member Functions | |
| static double | density (int nu, double x, double y, double rho) |
Computes the standard bivariate Student’s \(t\) density function ( pdf1bit ) with correlation \(\rho\) = rho and \(\nu\) = nu degrees of freedom. | |
| static double | cdf (int nu, double x, double y, double rho) |
Computes the standard bivariate Student’s \(t\) distribution ( cdf1bit ) using the method described in [65] . | |
| static double | barF (int nu, double x, double y, double rho) |
Computes the standard upper bivariate Student’s \(t\) distribution ( cdf2bit ). | |
| static double[] | getMean (int nu, double rho) |
| Returns the mean vector \(E[X] = (0, 0)\) of the bivariate Student’s. | |
| static double[][] | getCovariance (int nu, double rho) |
| Returns the covariance matrix of the bivariate Student’s \(t\) distribution. | |
| static double[][] | getCorrelation (int nu, double rho) |
| Returns the correlation matrix of the bivariate Student’s \(t\) distribution. | |
Protected Member Functions | |
| void | setParams (int nu, double rho) |
| Sets the parameters \(\nu\) = nu and \(\rho\) = rho of this object. | |
Additional Inherited Members | |
| Public Attributes inherited from umontreal.ssj.probdistmulti.ContinuousDistribution2Dim | |
| int | decPrec = 15 |
| Defines the target number of decimals of accuracy when approximating a distribution function, but there is no guarantee that this target is always attained. | |
Detailed Description
Extends the class ContinuousDistribution2Dim for the standard bivariate Student’s \(t\) distribution [94] (page 132).
The correlation between \(X\) and \(Y\) is \(\rho\) and the number of degrees of freedom is \(\nu\). Its probability density is
\[ f (x, y) = \frac{1}{2\pi\sqrt{1-\rho^2}}\left[1 + \frac{x^2 - 2\rho x y + y^2}{\nu(1-\rho^2)}\right]^{-(\nu+ 2)/2} , \tag{pdf1bit} \]
and the corresponding distribution function (the cdf) is
\[ T_{\nu}(x, y, \rho) = \frac{1}{2\pi\sqrt{1-\rho^2}} \int_{-\infty}^x dx \int_{-\infty}^y dy f (x, y). \tag{cdf1bit} \]
We also define the upper distribution function called barF as
\[ \overline{T}_{\nu}(x, y, \rho) = \frac{1}{2\pi\sqrt{1-\rho^2}} \int^{\infty}_x dx \int^{\infty}_y dy f(x,y). \tag{cdf2bit} \]
Definition at line 35 of file BiStudentDist.java.
Constructor & Destructor Documentation
◆ BiStudentDist()
| umontreal.ssj.probdistmulti.BiStudentDist.BiStudentDist | ( | int | nu, |
| double | rho ) |
Constructs a BiStudentDist object with correlation \(\rho= \) rho and \(\nu\) = nu degrees of freedom.
Definition at line 45 of file BiStudentDist.java.
Member Function Documentation
◆ barF() [1/2]
| double umontreal.ssj.probdistmulti.BiStudentDist.barF | ( | double | x, |
| double | y ) |
Computes the upper cumulative distribution function.
\(\overline{F}(x, y)\):
\[ \overline{F}(x, y) = P[X\ge x, Y \ge y] = \int^{\infty}_x ds \int^{\infty}_y dt f(s, t). \]
- 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 \((x, y)\)
Reimplemented from umontreal.ssj.probdistmulti.ContinuousDistribution2Dim.
Definition at line 60 of file BiStudentDist.java.
◆ barF() [2/2]
|
static |
Computes the standard upper bivariate Student’s \(t\) distribution ( cdf2bit ).
Definition at line 202 of file BiStudentDist.java.
◆ cdf() [1/2]
| double umontreal.ssj.probdistmulti.BiStudentDist.cdf | ( | double | x, |
| double | y ) |
Computes the distribution function \(F(x, y)\):
\[ F(x, y) = P[X\le x, Y \le y] = \int_{-\infty}^x ds \int_{-\infty}^y dt f(s, t). \]
- 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 \((x, y)\)
Reimplemented from umontreal.ssj.probdistmulti.ContinuousDistribution2Dim.
Definition at line 56 of file BiStudentDist.java.
◆ cdf() [2/2]
|
static |
Computes the standard bivariate Student’s \(t\) distribution ( cdf1bit ) using the method described in [65] .
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
\(\rho= \) rho and the number of degrees of freedom is \(\nu\) = nu.
Definition at line 92 of file BiStudentDist.java.
◆ density() [1/2]
| double umontreal.ssj.probdistmulti.BiStudentDist.density | ( | double | x, |
| double | y ) |
Returns \(f(x, y)\), the density of \((X, Y)\) evaluated at \((x, y)\).
- Parameters
-
x value \(x\) at which the density is evaluated y value \(y\) at which the density is evaluated
- Returns
- density function evaluated at \((x, y)\)
Reimplemented from umontreal.ssj.probdistmulti.ContinuousDistribution2Dim.
Definition at line 49 of file BiStudentDist.java.
◆ density() [2/2]
|
static |
Computes the standard bivariate Student’s \(t\) density function ( pdf1bit ) with correlation \(\rho\) = rho and \(\nu\) = nu degrees of freedom.
Definition at line 69 of file BiStudentDist.java.
◆ getCorrelation() [1/2]
| double[][] umontreal.ssj.probdistmulti.BiStudentDist.getCorrelation | ( | ) |
Returns the correlation matrix of the distribution, defined as.
\(\rho_{ij} = \sigma_{ij}/\sqrt{\sigma_{ii}\sigma_{jj}}\).
Reimplemented from umontreal.ssj.probdistmulti.ContinuousDistributionMulti.
Definition at line 261 of file BiStudentDist.java.
◆ getCorrelation() [2/2]
|
static |
Returns the correlation matrix of the bivariate Student’s \(t\) distribution.
Definition at line 269 of file BiStudentDist.java.
◆ getCovariance() [1/2]
| double[][] umontreal.ssj.probdistmulti.BiStudentDist.getCovariance | ( | ) |
Returns the variance-covariance matrix of the distribution, defined as
\(\sigma_{ij} = E[(X_i - \mu_i)(X_j - \mu_j)]\).
Reimplemented from umontreal.ssj.probdistmulti.ContinuousDistributionMulti.
Definition at line 235 of file BiStudentDist.java.
◆ getCovariance() [2/2]
|
static |
Returns the covariance matrix of the bivariate Student’s \(t\) distribution.
Definition at line 243 of file BiStudentDist.java.
◆ getMean() [1/2]
| double[] umontreal.ssj.probdistmulti.BiStudentDist.getMean | ( | ) |
Returns the mean vector of the distribution, defined as \(\mu_i = E[X_i]\).
Reimplemented from umontreal.ssj.probdistmulti.ContinuousDistributionMulti.
Definition at line 212 of file BiStudentDist.java.
◆ getMean() [2/2]
|
static |
Returns the mean vector \(E[X] = (0, 0)\) of the bivariate Student’s.
\(t\) distribution.
Definition at line 221 of file BiStudentDist.java.
◆ setParams()
|
protected |
Sets the parameters \(\nu\) = nu and \(\rho\) = rho of this object.
Definition at line 288 of file BiStudentDist.java.
The documentation for this class was generated from the following file:
- src/main/java/umontreal/ssj/probdistmulti/BiStudentDist.java
