NAG Library Function Document

nag_multi_students_t (g01hdc)

nag_multi_students_t (Integer n, const Nag_TailProbability tail[], const double a[], const double b[], double nu, const double delta[], Nag_Boolean iscov, double rc[], Integer pdrc, double epsabs, double epsrel, Integer numsub, Integer nsampl, Integer fmax, double *errest, NagError *fail)

3 Description

A random vector

x \in ℝ^{n}

that follows a Student's

t

-distribution with

ν

degrees of freedom and covariance matrix

Σ

has density:

\frac{Γ ((ν + n) / 2)}{Γ (ν / 2) ν^{n / 2} π^{n / 2} {|Σ|}^{1 / 2} {[1 + \frac{1}{ν} x^{T} Σ^{- 1} x]}^{(ν + n) / 2}},

and probability

p

given by:

p = \frac{Γ ((ν + n) / 2)}{Γ (ν / 2) \sqrt{|Σ| {(π ν)}^{n}}} \int_{a_{1}}^{b_{1}} \int_{a_{2}}^{b_{2}} \dots \int_{a_{n}}^{b_{n}} {(1 + x^{T} Σ^{- 1} x / ν)}^{- (ν + n) / 2} d x .

The method of calculation depends on the dimension

n

and degrees of freedom

ν

. The method of Dunnet and Sobel is used in the bivariate case if

ν

is a whole number. A Plackett transform followed by quadrature method is adopted in other bivariate cases and trivariate cases. In dimensions higher than three a number theoretic approach to evaluating multidimensional integrals is adopted.

Error estimates are supplied as the published accuracy in the Dunnet and Sobel case, a Monte–Carlo standard error for multidimensional integrals, and otherwise the quadrature error estimate.

A parameter

δ

allows for non-central probabilities. The number theoretic method is used if any

δ

is nonzero.

In cases other than the central bivariate with whole

ν

, nag_multi_students_t (g01hdc) attempts to evaluate probabilities within a requested accuracy

\max (ε_{a}, ε_{r} \times I)

, for an approximate integral value

I

, absolute accuracy

ε_{a}

and relative accuracy

ε_{r}

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 and Bretz F (2002) Methods for the computation of multivariate

t

-probabilities Journal of Computational and Graphical Statistics (11) 950–971

5 Arguments

1: n – IntegerInput

On entry:

n

, the number of dimensions.

Constraint:

1 < n \leq 1000

2: tail[n] – const Nag_TailProbabilityInput

On entry: defines the calculated probability, set

tail [i - 1]

to:

$tail [i - 1] = Nag_LowerTail$: If the $i$ th lower limit $a_{i}$ is negative infinity.
$tail [i - 1] = Nag_UpperTail$: If the $i$ th upper limit $b_{i}$ is infinity.
$tail [i - 1] = Nag_Central$: If both $a_{i}$ and $b_{i}$ are finite.

Constraint:

tail [i - 1] = Nag_LowerTail

Nag_UpperTail

Nag_Central

, for

i = 1, 2, \dots, n

3: a[n] – const doubleInput

On entry:

a_{i}

, for

i = 1, 2, \dots, n

, the lower integral limits of the calculation.

tail [i - 1] = Nag_LowerTail

a [i - 1]

is not referenced and the

i

th lower limit of integration is

- \infty

4: b[n] – const doubleInput

On entry:

b_{i}

, for

i = 1, 2, \dots, n

, the upper integral limits of the calculation.

tail [i - 1] = Nag_UpperTail

b [i - 1]

is not referenced and the

i

th upper limit of integration is

\infty

Constraint: if

tail [i - 1] = Nag_Central

b [i - 1] > a [i - 1]

5: nu – doubleInput

On entry:

ν

, the degrees of freedom.

Constraint:

nu > 0.0

6: delta[n] – const doubleInput

On entry:

delta [i - 1]

the noncentrality parameter for the

i

th dimension, for

i = 1, 2, \dots, n

; set

delta [i - 1] = 0

for the central probability.

7: iscov – Nag_BooleanInput

On entry: set

iscov = Nag_TRUE

if the covariance matrix is supplied and

iscov = Nag_FALSE

if the correlation matrix is supplied.

8: rc[ $n \times pdrc$ ] – doubleInput/Output

Note: the

(i, j)

th element of the matrix is stored in

rc [(i - 1) \times pdrc + j - 1]

On entry: the lower triangle of either the covariance matrix (if

iscov = Nag_TRUE

) or the correlation matrix (if

iscov = Nag_FALSE

). In either case the array elements corresponding to the upper triangle of the matrix need not be set.

On exit: the strict upper triangle of rc contains the correlation matrix used in the calculations.

9: pdrc – IntegerInput

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

Constraint:

pdrc \geq n

10: epsabs – doubleInput

On entry:

ε_{a}

, the absolute accuracy requested in the approximation. If epsabs is negative, the absolute value is used.

Suggested value:

0.0

11: epsrel – doubleInput

On entry:

ε_{r}

, the relative accuracy requested in the approximation. If epsrel is negative, the absolute value is used.

Suggested value:

0.001

12: numsub – IntegerInput

On entry: if quadrature is used, the number of sub-intervals used by the quadrature algorithm; otherwise numsub is not referenced.

Suggested value:

350

Constraint: if referenced,

numsub > 0

13: nsampl – IntegerInput

On entry: if quadrature is used, nsampl is not referenced; otherwise nsampl is the number of samples used to estimate the error in the approximation.

Suggested value:

8

Constraint: if referenced,

nsampl > 0

14: fmax – IntegerInput

On entry: if a number theoretic approach is used, the maximum number of evaluations for each integrand function.

Suggested value:

1000 \times n

Constraint: if referenced,

fmax \geq 1

15: errest – double *Output

On exit: an estimate of the error in the calculated probability.

16: fail – NagError *Input/Output

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

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_ARRAY_SIZE: On entry, $pdrc = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $pdrc \geq n$ .
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $fmax = ⟨value⟩$ .
Constraint: $fmax \geq 1$ .
On entry, $n = ⟨value⟩$ .
Constraint: $1 < n \leq 1000$ .
On entry, $nsampl = ⟨value⟩$ .
Constraint: $nsampl \geq 1$ .
On entry, $numsub = ⟨value⟩$ .
Constraint: $numsub \geq 1$ .
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_INVALID_ARRAY: On entry, the information supplied in rc is invalid.
NE_REAL: On entry, $nu = ⟨value⟩$ .
Constraint: degrees of freedom $nu > 0.0$ .
NE_REAL_2: On entry, $k = ⟨value⟩$ .
Constraint: $b [k - 1] > a [k - 1]$ for a central probability.

7 Accuracy

An estimate of the error in the calculation is given by the value of errest on exit.

8 Parallelism and Performance

nag_multi_students_t (g01hdc) is not threaded by NAG in any implementation.

nag_multi_students_t (g01hdc) 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

None.

10 Example

This example prints two probabilities from the Student's

t

-distribution.

NAG Library Function Documentnag_multi_students_t (g01hdc)

+− 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_students_t (g01hdc)