Real (Kind=nag_wp)	::	g01hbf
Integer, Intent (In)	::	n, ldsig, lwk
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	a(n), b(n), xmu(n), sig(ldsig,n), tol
Real (Kind=nag_wp), Intent (Out)	::	wk(lwk)
Character (1), Intent (In)	::	tail

C Header Interface

#include <nag.h>

double

g01hbf_ (const char *tail, const Integer *n, const double a[], const double b[], const double xmu[], const double sig[], const Integer *ldsig, const double *tol, double wk[], const Integer *lwk, Integer *ifail, const Charlen length_tail)

The routine may be called by the names g01hbf or nagf_stat_prob_multi_normal.

3 Description

Let the vector random variable

X = {(X_{1}, X_{2}, \dots, X_{n})}^{T}

follow an

n

-dimensional multivariate Normal distribution with mean vector

μ

and

n \times n

variance-covariance matrix

Σ

, then the probability density function,

f (X : μ, Σ)

, is given by

f (X : μ, Σ) = {(2 π)}^{- (1 / 2) n} {| Σ |}^{- 1 / 2} \exp (- \frac{1}{2} {(X - μ)}^{T} Σ^{- 1} (X - μ)) .

The lower tail probability is defined by:

P (X_{1} \leq b_{1}, \dots, X_{n} \leq b_{n} : μ, Σ) = \int_{- \infty}^{b_{1}} \dots \int_{- \infty}^{b_{n}} f (X : μ, Σ) d X_{n} \dots d X_{1} .

The upper tail probability is defined by:

P (X_{1} \geq a_{1}, \dots, X_{n} \geq a_{n} : μ, Σ) = \int_{a_{1}}^{\infty} \dots \int_{a_{n}}^{\infty} f (X : μ, Σ) d X_{n} \dots d X_{1} .

The central probability is defined by:

P (a_{1} \leq X_{1} \leq b_{1}, \dots, a_{n} \leq X_{n} \leq b_{n} : μ, Σ) = \int_{a_{1}}^{b_{1}} \dots \int_{a_{n}}^{b_{n}} f (X : μ, Σ) d X_{n} \dots d X_{1} .

To evaluate the probability for

n \geq 3

, the probability density function of

X_{1}, X_{2}, \dots, X_{n}

is considered as the product of the conditional probability of

X_{1}, X_{2}, \dots, X_{n - 2}

given

X_{n - 1}

and

X_{n}

and the marginal bivariate Normal distribution of

X_{n - 1}

and

X_{n}

. The bivariate Normal probability can be evaluated as described in g01haf and numerical integration is then used over the remaining

n - 2

dimensions. In the case of

n = 3

, d01ajf is used and for

n > 3

d01fcf is used.

To evaluate the probability for

n = 1

a direct call to g01eaf is made and for

n = 2

calls to g01haf are made.

4 References

Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin

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: $n$ – Integer Input

On entry:

n

, the number of dimensions.

Constraint:

1 \leq n \leq 10

3: $a (n)$ – Real (Kind=nag_wp) array Input

On entry: if

tail ='C'

'U'

, the lower bounds,

a_{i}

, for

i = 1, 2, \dots, n

tail ='L'

, a is not referenced.

4: $b (n)$ – Real (Kind=nag_wp) array Input

On entry: if

tail ='C'

'L'

, the upper bounds,

b_{i}

, for

i = 1, 2, \dots, n

tail ='U'

, b is not referenced.

Constraint: if

tail ='C'

a (i) < b (i)

, for

i = 1, 2, \dots, n

5: $xmu (n)$ – Real (Kind=nag_wp) array Input

On entry:

μ

, the mean vector of the multivariate Normal distribution.

6: $sig (ldsig, n)$ – Real (Kind=nag_wp) array Input

On entry:

Σ

, the variance-covariance matrix of the multivariate Normal distribution. Only the lower triangle is referenced.

Constraint:

Σ

must be positive definite.

7: $ldsig$ – Integer Input

On entry: the first dimension of the array sig as declared in the (sub)program from which g01hbf is called.

Constraint:

ldsig \geq n

8: $tol$ – Real (Kind=nag_wp) Input

On entry: if

n > 2

the relative accuracy required for the probability, and if the upper or the lower tail probability is requested then tol is also used to determine the cut-off points, see Section 7.

n = 1

, tol is not referenced.

Suggested value:

tol = 0.0001

Constraint: if

n > 1

tol > 0.0

9: $wk (lwk)$ – Real (Kind=nag_wp) array Workspace

10: $lwk$ – Integer Input

On entry: the length of workspace provided in array wk. This workspace is used by the numerical integration routines d01ajf for

n = 3

and d01fcf for

n > 3

n = 3

, the maximum number of sub-intervals used by d01ajf is

lwk / 4

. Note, however, increasing lwk above

1000

will not increase the maximum number of sub-intervals above

250

n > 3

the maximum number of integrand evaluations used by d01fcf is

α (lwk / n - 1)

, where

α = 2^{n - 2} + 2 {(n - 2)}^{2} + 2 (n - 2) + 1

n = 1

2

, wk will not be used.

Suggested value:

2000

n > 3

and

1000

n = 3

Constraints:

if $n \leq 2$ , $lwk \geq 1$ ;
if $n \geq 3$ , $lwk \geq 4 \times n$ .

11: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

−1

is recommended since useful values can be provided in some output arguments even when

ifail \neq 0

on exit. 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:

Note: in some cases g01hbf may return useful information.

If on exit

ifail = 1

2

3

, then g01hbf returns zero.

$ifail = 1$: On entry, $ldsig = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $ldsig \geq n$ .

On entry, $lwk = ⟨ value ⟩$ .
Constraint: if $n \leq 2$ , $lwk \geq 1$ .

On entry, $lwk = ⟨ value ⟩$ .
Constraint: if $n \geq 3$ , $lwk \geq 4 \times n$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $1 \leq n \leq 10$ .

On entry, $tail = ⟨ value ⟩$ .
Constraint: $tail ='L'$ , $'U'$ or $'C'$ .

On entry, $tol = ⟨ value ⟩$ .
Constraint: $tol > 0.0$ .

$ifail = 2$: On entry, the $⟨ value ⟩$ value in b is less than or equal to the corresponding value in a.

$ifail = 3$: On entry, sig is not positive definite.

$ifail = 4$: Full accuracy not achieved, relative accuracy $= ⟨ value ⟩$ . A larger value of tol can be tried or the length of the workspace increased. The returned value is an approximation to the required result.

$ifail = 5$: Accuracy requested by tol is too strict: $tol = ⟨ value ⟩$ . Round-off error has prevented the requested accuracy from being achieved; a larger value of tol can be tried. The returned value will be an approximation to the required result.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The accuracy should be as specified by tol. When on exit

ifail = 4

the approximate accuracy achieved is given in the error message. For the upper and lower tail probabilities the infinite limits are approximated by cut-off points for the

n - 2

dimensions over which the numerical integration takes place; these cut-off points are given by

Φ^{- 1} (tol / (10 \times n))

, where

Φ^{- 1}

is the inverse univariate Normal distribution function.

8 Parallelism and Performance

g01hbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g01hbf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The time taken is related to the number of dimensions, the range over which the integration takes place (

b_{i} - a_{i}

, for

i = 1, 2, \dots, n

) and the value of

Σ

as well as the accuracy required. As the numerical integration does not take place over the last two dimensions speed may be improved by arranging

X

so that the largest ranges of integration are for

X_{n - 1}

and

X_{n}

10 Example

This example reads in the mean and covariance matrix for a multivariate Normal distribution and computes and prints the associated central probability.

g01hb: FL CL CPP AD

NAG FL Interfaceg01hbf (prob_​multi_​normal)

▸▿ 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 FL Interface
g01hbf (prob_multi_normal)