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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 2.7 in How to Use the NAG Library and its Documentation).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
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

nag_bivariate_students_t (g01hcc) is not threaded in any implementation.

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)