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NAG Toolbox: nag_stat_prob_bivariate_normal (g01ha)

Purpose

nag_stat_prob_bivariate_normal (g01ha) returns the lower tail probability for the bivariate Normal distribution.

Syntax

[result, ifail] = g01ha(x, y, rho)
[result, ifail] = nag_stat_prob_bivariate_normal(x, y, rho)

Description

For the two random variables $\left(X,Y\right)$ following a bivariate Normal distribution with
 $EX=0, EY=0, EX2=1, EY2=1 and EXY=ρ,$
the lower tail probability is defined by:
 $PX≤x,Y≤y:ρ=12π⁢1-ρ2 ∫-∞y ∫-∞x exp- X2- 2ρ XY+Y2 21-ρ2 dXdY.$
For a more detailed description of the bivariate Normal distribution and its properties see Abramowitz and Stegun (1972) and Kendall and Stuart (1969). The method used is described by Genz (2004).

References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Genz A (2004) Numerical computation of rectangular bivariate and trivariate Normal and $t$ probabilities Statistics and Computing 14 151–160
Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin

Parameters

Compulsory Input Parameters

1:     $\mathrm{x}$ – double scalar
$x$, the first argument for which the bivariate Normal distribution function is to be evaluated.
2:     $\mathrm{y}$ – double scalar
$y$, the second argument for which the bivariate Normal distribution function is to be evaluated.
3:     $\mathrm{rho}$ – double scalar
$\rho$, the correlation coefficient.
Constraint: $-1.0\le {\mathbf{rho}}\le 1.0$.

None.

Output Parameters

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

Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{rho}}<-1.0$, or ${\mathbf{rho}}>1.0$.
If on exit ${\mathbf{ifail}}={\mathbf{1}}$ then nag_stat_prob_bivariate_normal (g01ha) returns zero.
${\mathbf{ifail}}=-99$
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

Accuracy of the hybrid algorithm implemented here is discussed in Genz (2004). This algorithm should give a maximum absolute error of less than $5×{10}^{-16}$.

The probabilities for the univariate Normal distribution can be computed using nag_specfun_cdf_normal (s15ab) and nag_specfun_compcdf_normal (s15ac).

Example

This example reads values of $x$ and $y$ for a bivariate Normal distribution along with the value of $\rho$ and computes the lower tail probabilities.
```function g01ha_example

fprintf('g01ha example results\n\n');

x   = [    1.7    0      3.3     9.1];
y   = [   23.1    0     11.1     9.1 ];
rho = [    0      0.1    0.54    0.17];
p   = x;

fprintf('     x       y      rho      p\n');
for j = 1:numel(p)
[p(j), ifail] = g01ha( ...
x(j), y(j), rho(j));
end

fprintf('%8.3f%8.3f%8.3f%8.4f\n', [x; y; rho; p]);

```
```g01ha example results

x       y      rho      p
1.700  23.100   0.000  0.9554
0.000   0.000   0.100  0.2659
3.300  11.100   0.540  0.9995
9.100   9.100   0.170  1.0000
```

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