# NAG Library Routine Document

## 1Purpose

g01hcf returns probabilities for the bivariate Student's $t$-distribution, via the routine name.

## 2Specification

Fortran Interface
 Function g01hcf ( tail, a, b, df, rho,
 Real (Kind=nag_wp) :: g01hcf Integer, Intent (In) :: df Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: a(2), b(2), rho Character (1), Intent (In) :: tail
#include nagmk26.h
 double g01hcf_ (const char *tail, const double a[], const double b[], const Integer *df, const double *rho, Integer *ifail, const Charlen length_tail)

## 3Description

Let the vector random variable $X={\left({X}_{1},{X}_{2}\right)}^{\mathrm{T}}$ follow a bivariate Student's $t$-distribution with degrees of freedom $\nu$ and correlation $\rho$, then the probability density function is given by
 $fX:ν,ρ = 1 2π 1-ρ2 1 + X12 + X22 - 2 ρ X1 X2 ν 1-ρ2 -ν/2-1 .$
The lower tail probability is defined by:
 $P X1 ≤ b1 , X2 ≤ b2 :ν,ρ = ∫ -∞ b1 ∫ -∞ b2 fX:ν,ρ dX2 dX1 .$
The upper tail probability is defined by:
 $P X1 ≥ a1 , X2 ≥ a2 :ν,ρ = ∫ a1 ∞ ∫ a2 ∞ fX:ν,ρ dX2 dX1 .$
The central probability is defined by:
 $P a1 ≤ X1 ≤ b1 , a2 ≤ X2 ≤ b2 :ν,ρ = ∫ a1 b1 ∫ a2 b2 fX:ν,ρ dX2 dX1 .$
Calculations use the Dunnet and Sobel (1954) method, as described by Genz (2004).

## 4References

Dunnet 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

## 5Arguments

1:     $\mathbf{tail}$ – Character(1)Input
On entry: indicates which probability is to be returned.
${\mathbf{tail}}=\text{'L'}$
The lower tail probability is returned.
${\mathbf{tail}}=\text{'U'}$
The upper tail probability is returned.
${\mathbf{tail}}=\text{'C'}$
The central probability is returned.
Constraint: ${\mathbf{tail}}=\text{'L'}$, $\text{'U'}$ or $\text{'C'}$.
2:     $\mathbf{a}\left(2\right)$ – Real (Kind=nag_wp) arrayInput
On entry: if ${\mathbf{tail}}=\text{'C'}$ or $\text{'U'}$, the lower bounds ${a}_{1}$ and ${a}_{2}$.
If ${\mathbf{tail}}=\text{'L'}$, a is not referenced.
3:     $\mathbf{b}\left(2\right)$ – Real (Kind=nag_wp) arrayInput
On entry: if ${\mathbf{tail}}=\text{'C'}$ or $\text{'L'}$, the upper bounds ${b}_{1}$ and ${b}_{2}$.
If ${\mathbf{tail}}=\text{'U'}$, b is not referenced.
Constraint: if ${\mathbf{tail}}=\text{'C'}$, ${a}_{i}<{b}_{i}$, for $\mathit{i}=1,2$.
4:     $\mathbf{df}$ – IntegerInput
On entry: $\nu$, the degrees of freedom of the bivariate Student's $t$-distribution.
Constraint: ${\mathbf{df}}\ge 1$.
5:     $\mathbf{rho}$ – Real (Kind=nag_wp)Input
On entry: $\rho$, the correlation of the bivariate Student's $t$-distribution.
Constraint: $-1.0\le {\mathbf{rho}}\le 1.0$.
6:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
If on exit, ${\mathbf{ifail}}\ne 0$, then g01hcf returns zero.
${\mathbf{ifail}}=1$
On entry, ${\mathbf{tail}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{tail}}=\text{'L'}$, $\text{'U'}$ or $\text{'C'}$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{b}}\left(i\right)\le {\mathbf{a}}\left(i\right)$ for central probability, for some $i=1,2$.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{df}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{df}}\ge 1$.
${\mathbf{ifail}}=5$
On entry, ${\mathbf{rho}}=〈\mathit{\text{value}}〉$.
Constraint: $-1.0\le {\mathbf{rho}}\le 1.0$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

Accuracy of the algorithm implemented here is discussed in comparison with algorithms based on a generalized Placket formula by Genz (2004), who recommends the Dunnet and Sobel method. This implementation should give a maximum absolute error of the order of ${10}^{-16}$.

## 8Parallelism and Performance

g01hcf is not threaded in any implementation.

None.

## 10Example

This example calculates the bivariate Student's $t$ probability given the choice of tail and degrees of freedom, correlation and bounds.

### 10.1Program Text

Program Text (g01hcfe.f90)

### 10.2Program Data

Program Data (g01hcfe.d)

### 10.3Program Results

Program Results (g01hcfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017