1: $df$ – int64int32nag_int scalar: $ν$ , the degrees of freedom of the bivariate Student's $t$ -distribution.

Constraint: $df \geq 1$ .
2: $rho$ – double scalar: $ρ$ , the correlation of the bivariate Student's $t$ -distribution.

Constraint: $- 1.0 \leq rho \leq 1.0$ .

Optional Input Parameters

1: $a (2)$ – double array: If upper tail or central probablilities are to be returned, a should supply the lower bounds, $a_{i}$ , for $i = 1, 2$ .
2: $b (2)$ – double array: If lower tail or central probablilities are to be returned, b should supply the upper bounds, $b_{i}$ , for $i = 1, 2$ .

Output Parameters

1: $result$ – double scalar: The result of the function.
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

If on exit,

ifail \neq 0

, then nag_stat_prob_bivariate_students_t (g01hc) returns zero.

$ifail = 3$: On entry, $b (i) \leq a (i)$ for central probability, for some $i = 1, 2$ .

$ifail = 4$: Constraint: $df \geq 1$ .

$ifail = 5$: Constraint: $- 1.0 \leq rho \leq 1.0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

Accuracy of the algorithm implemented here is discussed in comparison with algorithms based on a generalized Placket formula by Genz (2004), who recommends the Dunnet and Sobel method. This implementation should give a maximum absolute error of the order of

10^{- 16}

Further Comments

None.

Example

This example calculates the bivariate Student's

t

probability given the choice of tail and degrees of freedom, correlation and bounds.

Open in the MATLAB editor: g01hc_example

function g01hc_example


fprintf('g01hc example results\n\n');

a     = [-Inf    -Inf;
	  -40       0;
           -2       8];
b     = [   4       0.8;
	    2       4;
          Inf     Inf];
tail = {'L'; 'C'; 'U'};
df    = int64([   8;     12;      2   ]);
rho   = [   0.6;   -0.2;    0.3 ];
p     = rho;

fprintf('%9s%12s%12s%12s%7s%6s%6s%4s\n', 'a(1)', 'b(1)', 'a(2)', ...
	'b(2)', 'df', 'rho','tail', 'p');
for j = 1:numel(df)
  if tail{j}=='L'
    [p(j), ifail] = g01hc( ...
			   df(j), rho(j), 'b', b(j,:));
  elseif tail{j}=='C'
    [p(j), ifail] = g01hc( ...
			   df(j), rho(j), 'a', a(j,:), 'b', b(j,:));
  else
    [p(j), ifail] = g01hc( ...
			   df(j), rho(j), 'a', a(j,:));
  end
  fprintf('%12.4e%12.4e%12.4e%12.4e%4d%7.4f%5s%7.4f\n', a(j,1), b(j,1), ...
	  a(j,2), b(j,2), df(j), rho(j), tail{j}, p(j));
end

g01hc example results

     a(1)        b(1)        a(2)        b(2)     df   rho  tail   p
        -Inf  4.0000e+00        -Inf  8.0000e-01   8 0.6000    L 0.7764
 -4.0000e+01  2.0000e+00  0.0000e+00  4.0000e+00  12-0.2000    C 0.4876
 -2.0000e+00         Inf  8.0000e+00         Inf   2 0.3000    U 0.0059

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox