NAG Library Routine Document

G01HDF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

G01HDF returns a probability associated with a multivariate Student's

t

-distribution.

2 Specification

FUNCTION G01HDF (

N, TAIL, A, B, NU, DELTA, ISCOV, RC, LDRC, EPSABS, EPSREL, NUMSUB, NSAMPL, FMAX, ERREST, IFAIL)

REAL (KIND=nag_wp) G01HDF

INTEGER	N, ISCOV, LDRC, NUMSUB, NSAMPL, FMAX, IFAIL
REAL (KIND=nag_wp)	A(N), B(N), NU, DELTA(N), RC(LDRC,N), EPSABS, EPSREL, ERREST
CHARACTER(1)	TAIL(N)

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 intergrals, 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

, G01HDF 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 Parameters

1: N – INTEGERInput

On entry:

n

, the number of dimensions.

Constraint:

1 < N < 1000

2: TAIL(N) – CHARACTER(1) arrayInput

On entry: defines the calculated probability, set

TAIL (i)

to:

$TAIL (i) ='L'$: If the $i$ th lower limit $a_{i}$ is negative infinity.
$TAIL (i) ='U'$: If the $i$ th upper limit $b_{i}$ is infinity.
$TAIL (i) ='C'$: If both $a_{i}$ and $b_{i}$ are finite.

Constraint:

TAIL (i) ='L'

'U'

'C'

, for

i = 1, 2, \dots, N

3: A(N) – REAL (KIND=nag_wp) arrayInput

On entry:

a_{i}

, for

i = 1, 2, \dots, n

, the lower integral limits of the calculation.

TAIL (i) ='L'

A (i)

is not referenced and the

i

th lower limit of integration is

- \infty

4: B(N) – REAL (KIND=nag_wp) arrayInput

On entry:

b_{i}

, for

i = 1, 2, \dots, n

, the upper integral limits of the calculation.

TAIL (i) ='U'

B (i)

is not referenced and the

i

th upper limit of integration is

\infty

Constraint: if

TAIL (i) ='C'

B (i) > A (i)

5: NU – REAL (KIND=nag_wp)Input

On entry:

ν

, the degrees of freedom.

Constraint:

NU > 0.0

6: DELTA(N) – REAL (KIND=nag_wp) arrayInput

On entry:

DELTA (i)

the noncentrality parameter for the

i

th dimension, for

i = 1, 2, \dots, N

; set

DELTA (i) = 0

for the central probability.

7: ISCOV – INTEGERInput

On entry: set

ISCOV = 1

if the covariance matrix is supplied and

ISCOV = 2

if the correlation matrix is supplied.

Constraint:

ISCOV = 1

2

8: RC(LDRC,N) – REAL (KIND=nag_wp) arrayInput/Output

On entry: the lower triangle of the matrix must contain the covariance matrix if

ISCOV = 1

or the correlation matrix if

ISCOV = 2

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

9: LDRC – INTEGERInput

On entry: the first dimension of the array RC as declared in the (sub)program from which G01HDF is called.

Constraint:

LDRC \geq N

10: EPSABS – REAL (KIND=nag_wp)Input

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 – REAL (KIND=nag_wp)Input

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; 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 – REAL (KIND=nag_wp)Output

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

16: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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:

$IFAIL = 1$: On entry, $N = ⟨value⟩$ .
Constraint: $N \geq 2$ and $N \leq 1000$ .

$IFAIL = 2$: On entry, $TAIL (k) = "⟨value⟩"$ .
Constraint: $TAIL (k) ='L'$ , $'U'$ or $'C'$ .

$IFAIL = 4$: On entry, $k = ⟨value⟩$ .
Constraint: $B (k) \leq A (k)$ for a central probability.

$IFAIL = 5$: On entry, $NU = ⟨value⟩$ .
Constraint: degrees of freedom $NU > 0.0$ .

$IFAIL = 8$: On entry, $ISCOV = ⟨value⟩$ .
Constraint: $ISCOV = 1$ or $2$ .

$IFAIL = 9$: On entry, the information supplied in RC is invalid.

$IFAIL = 10$: On entry, $LDRC = ⟨value⟩$ and $N = ⟨value⟩$ .
Constraint: $LDRC \geq N$ .

$IFAIL = 12$: On entry, $NUMSUB = ⟨value⟩$ .
Constraint: $NUMSUB \geq 1$ .

$IFAIL = 13$: On entry, $NSAMPL = ⟨value⟩$ .
Constraint: $NSAMPL \geq 1$ .

$IFAIL = 14$: On entry, $FMAX = ⟨value⟩$ .
Constraint: $FMAX \geq 1$ .

7 Accuracy

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

8 Further Comments

None.

9 Example

This example prints two probabilities from the Student's

t

-distribution.

NAG Library Routine DocumentG01HDF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G01HDF