NAG Library Function Document

nag_bivariate_normal_dist (g01hac)

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

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 3.6 in the Essential Introduction).

On any of the error conditions listed below nag_bivariate_normal_dist (g01hac) returns $0.0$ .
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.
NE_REAL_ARG_GT: On entry, $rho = ⟨value⟩$ .
Constraint: $rho \leq 1.0$ .
NE_REAL_ARG_LT: On entry, $rho = ⟨value⟩$ .
Constraint: $rho \geq - 1.0$ .

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}

Not applicable.

The probabilities for the univariate Normal distribution can be computed using nag_cumul_normal (s15abc) and nag_cumul_normal_complem (s15acc).

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.