naginterfaces.library.stat.prob_​bivariate_​students_​t

naginterfaces.library.stat.prob_bivariate_students_t(df, rho, a=None, b=None)

prob_bivariate_students_t returns probabilities for the bivariate Student’s $t$-distribution.

For full information please refer to the NAG Library document for g01hc

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g01/g01hcf.html

Parameters

dfint

$\nu$, the degrees of freedom of the bivariate Student’s $t$-distribution.

rhofloat

$\rho$, the correlation of the bivariate Student’s $t$-distribution.

aNone or float, array-like, shape $\left(2\right)$, optional

If upper tail or central probablilities are to be returned, $a$ should supply the lower bounds, $a_{\text{i}}$, for $\text{i}=1,2,\dots ,2$.

bNone or float, array-like, shape $\left(2\right)$, optional

If lower tail or central probablilities are to be returned, $b$ should supply the upper bounds, $b_{\text{i}}$, for $\text{i}=1,2,\dots ,2$.

Returns

pfloat

The probabilities for the bivariate Student’s $t$-distribution.

Raises

NagValueError

(errno $1$)

On entry, $\text{tail}=⟨value⟩$.

Constraint: $\text{tail}=\text{'L'}$, $\text{'U'}$ or $\text{'C'}$.

(errno $3$)

On entry, $b[i-1]\le a[i-1]$ for central probability, for some $i=1,2$.

(errno $4$)

On entry, $df=⟨value⟩$.

Constraint: $df\ge 1$.

(errno $5$)

On entry, $rho=⟨value⟩$.

Constraint: $-1.0\le rho\le 1.0$.

Notes

Let the vector random variable $X={(X_1,X_2)}^T$ follow a bivariate Student’s $t$-distribution with degrees of freedom $\nu$ and correlation $\rho$, then the probability density function is given by

$$(X:\nu ,\rho )=\frac{1}{2\pi \sqrt{1-{\rho }^2}}{(1+\frac{X_1^2+X_2^2-2\rho X_1{X}_2}{\nu (1-{\rho }^2)})}^{-\nu /2-1}\text{.}$$

The lower tail probability is defined by:

$$(X_1\le b_1,X_2\le b_2:\nu ,\rho )={\int }_{-\infty }^{b_1}{\int }_{-\infty }^{b_2}(X:\nu ,\rho )d{X}_{2}d{X}_1\text{.}$$

The upper tail probability is defined by:

$$(X_1\ge a_1,X_2\ge a_2:\nu ,\rho )={\int }_{a_1}^{\infty }{\int }_{a_2}^{\infty }(X:\nu ,\rho )d{X}_{2}d{X}_1\text{.}$$

The central probability is defined by:

$$(a_1\le X_1\le b_1,a_2\le X_2\le b_2:\nu ,\rho )={\int }_{a_1}^{b_1}{\int }_{a_2}^{b_2}(X:\nu ,\rho )d{X}_{2}d{X}_1\text{.}$$

Calculations use the Dunnett and Sobel (1954) method, as described by Genz (2004).

References

Dunnett, C W and Sobel, M, 1954, A bivariate generalization of Student’s $t$ -distribution, with tables for certain special cases, Biometrika (41), 153–169

Genz, A, 2004, Numerical computation of rectangular bivariate and trivariate Normal and $t$ probabilities, Statistics and Computing (14), 151–160