NAG Library Function Document

nag_bivariate_students_t (g01hcc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_bivariate_students_t (g01hcc) returns probabilities for the bivariate Student's

t

-distribution.

2 Specification

#include <nag.h>

#include <nagg01.h>

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 = {(X_{1}, X_{2})}^{T}

follow a bivariate Student's

t

-distribution with degrees of freedom

ν

and correlation

ρ

, then the probability density function is given by

f (X : ν, ρ) = \frac{1}{2 π \sqrt{1 - ρ^{2}}} {(1 + \frac{X_{1}^{2} + X_{2}^{2} - 2 ρ X_{1} X_{2}}{ν (1 - ρ^{2})})}^{- ν / 2 - 1} .

The lower tail probability is defined by:

P (X_{1} \leq b_{1}, X_{2} \leq b_{2} : ν, ρ) = \int_{- \infty}^{b_{1}} \int_{- \infty}^{b_{2}} f (X : ν, ρ) d X_{2} d X_{1} .

The upper tail probability is defined by:

P (X_{1} \geq a_{1}, X_{2} \geq a_{2} : ν, ρ) = \int_{a_{1}}^{\infty} \int_{a_{2}}^{\infty} f (X : ν, ρ) d X_{2} d X_{1} .

The central probability is defined by:

P (a_{1} \leq X_{1} \leq b_{1}, a_{2} \leq X_{2} \leq b_{2} : ν, ρ) = \int_{a_{1}}^{b_{1}} \int_{a_{2}}^{b_{2}} f (X : ν, ρ) d X_{2} d X_{1} .

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 Biometrika 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: tail – Nag_TailProbabilityInput

On entry: indicates which probability is to be returned.

$tail = Nag_LowerTail$: The lower tail probability is returned.
$tail = Nag_UpperTail$: The upper tail probability is returned.
$tail = Nag_Central$: The central probability is returned.

Constraint:

tail = Nag_LowerTail

Nag_UpperTail

Nag_Central

2: a[ $2$ ] – const doubleInput

On entry: if

tail = Nag_Central

Nag_UpperTail

, the lower bounds

a_{1}

and

a_{2}

tail = Nag_LowerTail

, a is not referenced.

3: b[ $2$ ] – const doubleInput

On entry: if

tail = Nag_Central

Nag_LowerTail

, the upper bounds

b_{1}

and

b_{2}

tail = Nag_UpperTail

, b is not referenced.

Constraint: if

tail = Nag_Central

a_{i} < b_{i}

, for

i = 1, 2

4: df – IntegerInput

On entry:

ν

, the degrees of freedom of the bivariate Student's

t

-distribution.

Constraint:

df \geq 1

5: rho – doubleInput

On entry:

ρ

, the correlation of the bivariate Student's

t

-distribution.

Constraint:

- 1.0 \leq rho \leq 1.0

6: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $df = ⟨value⟩$ .
Constraint: $df \geq 1$ .
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, $rho = ⟨value⟩$ .
Constraint: $- 1.0 \leq rho \leq 1.0$ .
NE_REAL_2: On entry, $b [i - 1] \leq a [i - 1]$ 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 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}

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 Example

This example calculates the bivariate Student's

t

probability given the choice of tail and degrees of freedom, correlation and bounds.

NAG Library Function Documentnag_bivariate_students_t (g01hcc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_bivariate_students_t (g01hcc)