NAG Library Routine Document

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

C Header Interface

#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)

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: $tail$ – Character(1)Input

On entry: indicates which probability is to be returned.

$tail ='L'$: The lower tail probability is returned.
$tail ='U'$: The upper tail probability is returned.
$tail ='C'$: The central probability is returned.

Constraint:

tail ='L'

'U'

'C'

2: $a (2)$ – Real (Kind=nag_wp) arrayInput

On entry: if

tail ='C'

'U'

, the lower bounds

a_{1}

and

a_{2}

tail ='L'

, a is not referenced.

3: $b (2)$ – Real (Kind=nag_wp) arrayInput

On entry: if

tail ='C'

'L'

, the upper bounds

b_{1}

and

b_{2}

tail ='U'

, b is not referenced.

Constraint: if

tail ='C'

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: $rho$ – Real (Kind=nag_wp)Input

On entry:

ρ

, the correlation of the bivariate Student's

t

-distribution.

Constraint:

- 1.0 \leq rho \leq 1.0

6: $ifail$ – IntegerInput/Output

On entry: ifail must be set to

0

- 1 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 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 $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

If on entry

ifail = 0

- 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,

ifail \neq 0

, then g01hcf returns zero.

$ifail = 1$: On entry, $tail = 〈value〉$ .
Constraint: $tail ='L'$ , $'U'$ or $'C'$ .

$ifail = 3$: On entry, $b (i) \leq a (i)$ for central probability, for some $i = 1, 2$ .

$ifail = 4$: On entry, $df = 〈value〉$ .
Constraint: $df \geq 1$ .

$ifail = 5$: On entry, $rho = 〈value〉$ .
Constraint: $- 1.0 \leq rho \leq 1.0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

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

g01hcf 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 Routine Document

g01hcf (prob_bivariate_students_t)

▸▿ 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

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