g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_bivariate_students_t (g01hcc)

## 1  Purpose

nag_bivariate_students_t (g01hcc) returns probabilities for the bivariate Student's $t$-distribution.

## 2  Specification

 #include #include
 double nag_bivariate_students_t (Nag_TailProbability tail, const double a[], const double b[], Integer df, double rho, NagError *fail)

## 3  Description

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).

## 4  References

Dunnet C W and Sobel M (1954) A bivariate generalization of Student's $t$-distribution, with tables for certain special cases Bimoetrika 41 153–169
Genz A (2004) Numerical computation of rectangular bivariate and trivariate Normal and $t$ probabilities Statistics and Computing 14 151–160

## 5  Arguments

1:     tailNag_TailProbabilityInput
On entry: indicates which probability is to be returned.
${\mathbf{tail}}=\mathrm{Nag_LowerTail}$
The lower tail probability is returned.
${\mathbf{tail}}=\mathrm{Nag_UpperTail}$
The upper tail probability is returned.
${\mathbf{tail}}=\mathrm{Nag_Central}$
The central probability is returned.
Constraint: ${\mathbf{tail}}=\mathrm{Nag_LowerTail}$, $\mathrm{Nag_UpperTail}$ or $\mathrm{Nag_Central}$.
2:     a[$2$]const doubleInput
On entry: if ${\mathbf{tail}}=\mathrm{Nag_Central}$ or $\mathrm{Nag_UpperTail}$, the lower bounds ${a}_{1}$ and ${a}_{2}$.
If ${\mathbf{tail}}=\mathrm{Nag_LowerTail}$, a is not referenced.
3:     b[$2$]const doubleInput
On entry: if ${\mathbf{tail}}=\mathrm{Nag_Central}$ or $\mathrm{Nag_LowerTail}$, the upper bounds ${b}_{1}$ and ${b}_{2}$.
If ${\mathbf{tail}}=\mathrm{Nag_UpperTail}$, b is not referenced.
Constraint: if ${\mathbf{tail}}=\mathrm{Nag_Central}$, ${a}_{i}<{b}_{i}$, for $\mathit{i}=1,2$.
4:     dfIntegerInput
On entry: $\nu$, the degrees of freedom of the bivariate Student's $t$-distribution.
Constraint: ${\mathbf{df}}\ge 1$.
5:     rhodoubleInput
On entry: $\rho$, the correlation of the bivariate Student's $t$-distribution.
Constraint: $-1.0\le {\mathbf{rho}}\le 1.0$.
6:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{df}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{df}}\ge 1$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REAL
On entry, ${\mathbf{rho}}=〈\mathit{\text{value}}〉$.
Constraint: $-1.0\le {\mathbf{rho}}\le 1.0$.
NE_REAL_2
On entry, ${\mathbf{b}}\left[i-1\right]\le {\mathbf{a}}\left[i-1\right]$ for central probability, for some $i=1,2$.

## 7  Accuracy

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

None.

## 9  Example

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

### 9.1  Program Text

Program Text (g01hcce.c)

### 9.2  Program Data

Program Data (g01hcce.d)

### 9.3  Program Results

Program Results (g01hcce.r)