NAG Library Routine Document

g01haf (prob_bivariate_normal)

1
Purpose

g01haf returns the lower tail probability for the bivariate Normal distribution.

2
Specification

Fortran Interface
Function g01haf ( x, y, rho, ifail)
Real (Kind=nag_wp):: g01haf
Integer, Intent (Inout):: ifail
Real (Kind=nag_wp), Intent (In):: x, y, rho
C Header Interface
#include <nagmk26.h>
double  g01haf_ (const double *x, const double *y, const double *rho, Integer *ifail)

3
Description

For the two random variables X,Y following a bivariate Normal distribution with
EX=0,  EY=0,  EX2=1,  EY2=1  and  EXY=ρ,  
the lower tail probability is defined by:
PXx,Yy:ρ=12π1-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).

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: -1.0rho1.0.
4:     ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1 or 1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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 or -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-1.0.
On entry, rho=value.
Constraint: rho1.0.
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
ifail=-999
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation 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 5×10-16.

8
Parallelism and Performance

g01haf is not threaded in any implementation.

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.

10.1
Program Text

Program Text (g01hafe.f90)

10.2
Program Data

Program Data (g01hafe.d)

10.3
Program Results

Program Results (g01hafe.r)