NAG Library Function Document

nag_multi_normal (g01hbc)

+− 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_multi_normal (g01hbc) returns the upper tail, lower tail or central probability associated with a multivariate Normal distribution of up to ten dimensions.

2 Specification

#include <nag.h>

#include <nagg01.h>

double

nag_multi_normal (Nag_TailProbability tail, Integer n, const double a[], const double b[], const double mean[], const double sigma[], Integer tdsig, double tol, Integer maxpts, NagError *fail)

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

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 nag_bivariate_normal_dist (g01hac) and numerical integration is then used over the remaining

n - 2

dimensions. In the case of

n = 3

, nag_1d_quad_gen_1 (d01sjc) is used and for

n > 3

nag_multid_quad_adapt_1 (d01wcc) is used.

To evaluate the probability for

n = 1

a direct call to nag_prob_normal (g01eac) is made and for

n = 2

calls to nag_bivariate_normal_dist (g01hac) 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 – 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: n – IntegerInput

On entry:

n

, the number of dimensions.

Constraint:

1 \leq n \leq 10

3: a[n] – const doubleInput

On entry: if

tail = Nag_Central

Nag_UpperTail

, the lower bounds,

a_{i}

, for

i = 1, 2, \dots, n

tail = Nag_LowerTail

, a is not referenced.

4: b[n] – const doubleInput

On entry: if

tail = Nag_Central

Nag_LowerTail

, the upper bounds,

b_{i}

, for

i = 1, 2, \dots, n

tail = Nag_UpperTail

b, is not referenced.

Constraint: if

tail = Nag_Central

a [i - 1] < b [i - 1]

, for

i = 1, 2, \dots, n

5: mean[n] – const doubleInput

On entry:

μ

, the mean vector of the multivariate Normal distribution.

6: sigma[ $n \times tdsig$ ] – const doubleInput

Note: the

(i, j)

th element of the matrix is stored in

sigma [(i - 1) \times tdsig + j - 1]

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: tdsig – IntegerInput

On entry: the stride separating matrix column elements in the array sigma.

Constraint:

tdsig \geq n

8: tol – doubleInput

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: maxpts – IntegerInput

On entry: the maximum number of sub-intervals or integrand evaluations.

n = 3

, then the maximum number of sub-intervals used by nag_1d_quad_gen_1 (d01sjc) is maxpts/4. Note however increasing maxpts above 1000 will not increase the maximum number of sub-intervals above 250.

n > 3

the maximum number of integrand evaluations used by nag_multid_quad_adapt_1 (d01wcc) is

α

(maxpts/

n - 1

), where

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

n = 1

or 2, then maxpts will not be used.

Suggested value: 2000 if

n > 3

and 1000 if

n = 3

Constraint: if

n \geq 3

maxpts \geq 4 \times n

10: fail – NagError *Input/Output

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

6 Error Indicators and Warnings

NE_2_INT_ARG_LT: On entry, $tdsig = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $tdsig \geq n$ .
NE_2_REAL_ARRAYS_CONS: On entry, the $⟨value⟩$ value in b is less than or equal to the corresponding value in a.
NE_ACC: Full accuracy not achieved, relative accuracy $= ⟨value⟩$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT_ARG_CONS: On entry, $maxpts = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: if $n \geq 3$ , $maxpts \geq 4 \times n$ .
On entry, $n = ⟨value⟩$ .
Constraint: $1 \leq n \leq 10$ .
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_POS_DEF: On entry, sigma is not positive definite.
NE_REAL_ARG_CONS: On entry, $tol = ⟨value⟩$ .
Constraint: $tol > 0.0$ .
NE_ROUND_OFF: Accuracy requested by tol is too strict: $tol = ⟨value⟩$ .

7 Accuracy

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

fail . code =

NE_ACC 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

nag_multi_normal (g01hbc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_multi_normal (g01hbc) 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 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.

NAG Library Function Documentnag_multi_normal (g01hbc)

+− 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_multi_normal (g01hbc)