NAG CL Interface
g01hac (prob_​bivariate_​normal)

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1 Purpose

g01hac returns the lower tail probability for the bivariate Normal distribution.

2 Specification

#include <nag.h>
double  g01hac (double x, double y, double rho, NagError *fail)
The function may be called by the names: g01hac, nag_stat_prob_bivariate_normal or nag_bivariate_normal_dist.

3 Description

For the two random variables (X,Y) following a bivariate Normal distribution with
E[X]=0,  E[Y]=0,  E[X2]=1,  E[Y2]=1  and  E[XY]=ρ,  
the lower tail probability is defined by:
P(Xx,Yy:ρ)=12π1-ρ2 -y -x exp(- (X2-2ρXY+Y2) 2(1-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 double Input
On entry: x, the first argument for which the bivariate Normal distribution function is to be evaluated.
2: y double Input
On entry: y, the second argument for which the bivariate Normal distribution function is to be evaluated.
3: rho double Input
On entry: ρ, the correlation coefficient.
Constraint: -1.0rho1.0.
4: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

On any of the error conditions listed below g01hac returns 0.0 .
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL_ARG_GT
On entry, rho=value.
Constraint: rho1.0.
NE_REAL_ARG_LT
On entry, rho=value.
Constraint: rho-1.0.

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

Background information to multithreading can be found in the Multithreading documentation.
g01hac 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 function. 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 s15abc and s15acc.

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 (g01hace.c)

10.2 Program Data

Program Data (g01hace.d)

10.3 Program Results

Program Results (g01hace.r)