NAG Toolbox: nag_multi_students_t (g01hd)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{tail}\left(\mathbf{n}\right)$ – cell array of strings

Defines the calculated probability, set $\mathbf{tail}\left(i\right)$ to:

$\mathbf{tail}\left(i\right)=\text{'L'}$

If the $i$th lower limit $a_i$ is negative infinity.

$\mathbf{tail}\left(i\right)=\text{'U'}$

If the $i$th upper limit $b_i$ is infinity.

$\mathbf{tail}\left(i\right)=\text{'C'}$

If both $a_i$ and $b_i$ are finite.

Constraint: $\mathbf{tail}\left(\mathit{i}\right)=\text{'L'}$, $\text{'U'}$ or $\text{'C'}$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$.

2:     $\mathrm{a}\left(\mathbf{n}\right)$ – double array

$a_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, the lower integral limits of the calculation.

If $\mathbf{tail}\left(i\right)=\text{'L'}$, $\mathbf{a}\left(i\right)$ is not referenced and the $i$th lower limit of integration is $-\infty$.

3:     $\mathrm{b}\left(\mathbf{n}\right)$ – double array

$b_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, the upper integral limits of the calculation.

If $\mathbf{tail}\left(i\right)=\text{'U'}$, $\mathbf{b}\left(i\right)$ is not referenced and the $i$th upper limit of integration is $\infty$.

Constraint: if $\mathbf{tail}\left(i\right)=\text{'C'}$, $\mathbf{b}\left(i\right)>\mathbf{a}\left(i\right)$.

4:     $\mathrm{nu}$ – double scalar

$\nu$, the degrees of freedom.

Constraint: $\mathbf{nu}>0.0$.

5:     $\mathrm{delta}\left(\mathbf{n}\right)$ – double array

$\mathbf{delta}\left(\mathit{i}\right)$ the noncentrality parameter for the $\mathit{i}$th dimension, for $\mathit{i}=1,2,\dots ,\mathbf{n}$; set $\mathbf{delta}\left(i\right)=0$ for the central probability.

6:     $\mathrm{iscov}$ – int64int32nag_int scalar

Set $\mathbf{iscov}=1$ if the covariance matrix is supplied and $\mathbf{iscov}=2$ if the correlation matrix is supplied.

Constraint: $\mathbf{iscov}=1$ or $2$.

7:     $\mathrm{rc}\left(\mathit{ldrc},\mathbf{n}\right)$ – double array

ldrc, the first dimension of the array, must satisfy the constraint $\mathit{ldrc}\ge \mathbf{n}$.

The lower triangle of either the covariance matrix (if $\mathbf{iscov}=1$) or the correlation matrix (if $\mathbf{iscov}=2$). In either case the array elements corresponding to the upper triangle of the matrix need not be set.

Optional Input Parameters

1:     $\mathrm{n}$ – int64int32nag_int scalar

Default: the dimension of the arrays tail, a, b, delta and the first dimension of the array rc and the second dimension of the array rc. (An error is raised if these dimensions are not equal.)

$n$, the number of dimensions.

Constraint: $1<\mathbf{n}\le 1000$.

2:     $\mathrm{epsabs}$ – double scalar

Default: $0.0$

${\epsilon }_a$, the absolute accuracy requested in the approximation. If epsabs is negative, the absolute value is used.

3:     $\mathrm{epsrel}$ – double scalar

Default: $0.001$

${\epsilon }_r$, the relative accuracy requested in the approximation. If epsrel is negative, the absolute value is used.

4:     $\mathrm{numsub}$ – int64int32nag_int scalar

Default: $350$

If quadrature is used, the number of sub-intervals used by the quadrature algorithm; otherwise numsub is not referenced.

Constraint: if referenced, $\mathbf{numsub}>0$.

5:     $\mathrm{nsampl}$ – int64int32nag_int scalar

Default: $8$

If quadrature is used, nsampl is not referenced; otherwise nsampl is the number of samples used to estimate the error in the approximation.

Constraint: if referenced, $\mathbf{nsampl}>0$.

6:     $\mathrm{fmax}$ – int64int32nag_int scalar

Default: $1000\times \mathbf{n}$

If a number theoretic approach is used, the maximum number of evaluations for each integrand function.

Constraint: if referenced, $\mathbf{fmax}\ge 1$.

Output Parameters

1:     $\mathrm{result}$ – double scalar

The result of the function.

2:     $\mathrm{rc}\left(\mathit{ldrc},\mathbf{n}\right)$ – double array

The strict upper triangle of rc contains the correlation matrix used in the calculations.

3:     $\mathrm{errest}$ – double scalar

An estimate of the error in the calculated probability.

4:     $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   $\mathbf{ifail}=1$

Constraint: $1<\mathbf{n}\le 1000$.

   $\mathbf{ifail}=2$

Constraint: $\mathbf{tail}\left(\mathit{k}\right)=\text{'L'}$, $\text{'U'}$ or $\text{'C'}$.

   $\mathbf{ifail}=4$

Constraint: $\mathbf{b}\left(k\right)>\mathbf{a}\left(k\right)$ for a central probability.

   $\mathbf{ifail}=5$

Constraint: degrees of freedom $\mathbf{nu}>0.0$.

   $\mathbf{ifail}=8$

Constraint: $\mathbf{iscov}=1$ or $2$.

   $\mathbf{ifail}=9$

On entry, the information supplied in rc is invalid.

   $\mathbf{ifail}=10$

Constraint: $\mathit{ldrc}\ge \mathbf{n}$.

   $\mathbf{ifail}=12$

Constraint: $\mathbf{numsub}\ge 1$.

   $\mathbf{ifail}=13$

Constraint: $\mathbf{nsampl}\ge 1$.

   $\mathbf{ifail}=14$

Constraint: $\mathbf{fmax}\ge 1$.

   $\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

   $\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

   $\mathbf{ifail}=-999$

Dynamic memory allocation failed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: g01hd_example

function g01hd_example


fprintf('g01hd example results\n\n');

iscov  = int64(1);

% Example 1
nu    = 10;
tail  = {'u';  'u';  'u';  'u';  'u'};
a     = [-0.1; -0.1; -0.1; -0.1; -0.1];
b     = [888;  888;  888;  888;  888];
delta = [0;    0;    0;    0;    0];
rc    = [1.00, 0.75, 0.75, 0.75, 0.75;
         0.75, 1.00, 0.75, 0.75, 0.75;
         0.75, 0.75, 1.00, 0.75, 0.75;
         0.75, 0.75, 0.75, 1.00, 0.75;
         0.75, 0.75, 0.75, 0.75, 1.00];

% Calculate probability
[result, rc, errest, ifail] = ...
  g01hd( ...
	 tail, a, b, nu, delta, iscov, rc);

fprintf('Example 1:\n');
fprintf('Probability:   %24.5e\nError estimate:%24.2e\n', result, errest);

% Example 2
nu = 3;
tail = {'l';   'l';  'l';  'l';  'l'};
a     = [888;  888;  888;  888;  888];
b     = [-0.1; -0.1; -0.1; -0.1; -0.1];
delta = [1;    2;    3;    3;    3];
rc    = [1.00, 0.75, 0.75, 0.75, 0.75;
         0.75, 1.00, 0.75, 0.75, 0.75;
         0.75, 0.75, 1.00, 0.75, 0.75;
         0.75, 0.75, 0.75, 1.00, 0.75;
         0.75, 0.75, 0.75, 0.75, 1.00];

% Calculate probability
[result, rc, errest, ifail] = ...
  g01hd( ...
	 tail, a, b, nu, delta, iscov, rc);

fprintf('\nExample 2:\n');
fprintf('Probability:   %24.5e\nError estimate:%24.2e\n', result, errest);

g01hd example results

Example 1:
Probability:                3.01642e-01
Error estimate:                1.09e-05

Example 2:
Probability:                8.62903e-05
Error estimate:                1.62e-07