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: $x$ – doubleInput: On entry: $x$ , the first argument for which the bivariate Normal distribution function is to be evaluated.
2: $y$ – doubleInput: On entry: $y$ , the second argument for which the bivariate Normal distribution function is to be evaluated.
3: $rho$ – doubleInput: On entry: $ρ$ , the correlation coefficient.

Constraint: $- 1.0 \leq rho \leq 1.0$ .
4: $fail$ – NagError *Input/Output: The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

On any of the error conditions listed below nag_bivariate_normal_dist (g01hac) returns $0.0$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL_ARG_GT: On entry, $rho = 〈value〉$ .
Constraint: $rho \leq 1.0$ .
NE_REAL_ARG_LT: On entry, $rho = 〈value〉$ .
Constraint: $rho \geq - 1.0$ .

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 \times 10^{- 16}

8

Parallelism and Performance

nag_bivariate_normal_dist (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 nag_cumul_normal (s15abc) and nag_cumul_normal_complem (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.

NAG Library Function Document

nag_bivariate_normal_dist (g01hac)

▸▿ 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

1

Purpose

2

Specification

3

Description