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).

4 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

5 Arguments

1: $x$ – Real (Kind=nag_wp) Input: On entry: $x$ , the first argument for which the bivariate Normal distribution function is to be evaluated.
2: $y$ – Real (Kind=nag_wp) Input: On entry: $y$ , the second argument for which the bivariate Normal distribution function is to be evaluated.
3: $rho$ – Real (Kind=nag_wp) Input: On entry: $ρ$ , the correlation coefficient.

Constraint: $- 1.0 \leq rho \leq 1.0$ .
4: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $−1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $−1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. 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, $rho = ⟨ value ⟩$ .
Constraint: $rho \geq - 1.0$ .

On entry, $rho = ⟨ value ⟩$ .
Constraint: $rho \leq 1.0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 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 \times 10^{−16}

8 Parallelism and Performance

g01haf 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 X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The probabilities for the univariate Normal distribution can be computed using s15abf and s15acf.

10 Example

This example reads values of

x

and

y

for a bivariate Normal distribution along with the value of

ρ

and computes the lower tail probabilities.

g01ha: FL CL CPP AD

NAG FL Interfaceg01haf (prob_​bivariate_​normal)

▸▿ 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 FL Interface
g01haf (prob_bivariate_normal)