approxbvncdf: APPROXIMATION OF BIVARIATE STANDARD NORMAL DISTRIBUTION

In weightedScores: Weighted Scores Method for Regression Models with Dependent Data

APPROXIMATION OF BIVARIATE STANDARD NORMAL DISTRIBUTION

Description

Approximation of bivariate standard normal cumulative distribution function (Johnson and Kotz, 1972).

Usage

approxbvncdf(r,x1,x2,x1s,x2s,x1c,x2c,x1f,x2f,t1,t2)

Arguments

r	The correlation parameter of bivariate standard normal distribution.
x1	x_1, see details.
x2	x_2, see details.
x1s	x_1^2.
x2s	x_2^2.
x1c	x_1^3.
x2c	x_2^3.
x1f	x_1^4.
x2f	x_2^4.
t1	\Phi(x_1)\Phi(x_2), where \Phi(\cdot) is the cdf of univariate standard normal distribution.
t2	\phi(x_1)\phi(x_2), where \phi(\cdot) is the density of univariate stamdard normal distribution.

Details

The approximation for the bivariate normal cdf is from Johnson and Kotz (1972), page 118. Let \Phi_2(x_1,x_2;\rho)=Pr(Z_1\le x_1,\,Z_2\le x_2), where (Z_1,Z_2) is bivariate normal with means 0, variances 1 and correlation \rho. An expansion, due to Pearson (1901), is

\Phi_2(x_1,x_2;\rho) =\Phi(x_1)\Phi(x_2) +\phi(x_1)\phi(x_2) \sum_{j=1}^\infty \rho^j \psi_j(x_1) \psi_j(x_2)/j!

where

\psi_j(z) = (-1)^{j-1} d^{j-1} \phi(z)/dz^{j-1}.

Since

\phi'(z) = -z\phi(z), \phi''(z) = (z^2-1)\phi(z) , \phi'''(z) = [2z-z(z^2-1)]\phi(z) = (3z-z^3)\phi(z) ,

\phi^{(4)}(z) = [3-3z^2-z(3z-z^3)]\phi(z) = (3-6z^2+z^4)\phi(z)

we have

\Phi_2(x_1,x_2;\rho) = \Phi(x_1)\Phi(x_2)+\phi(x_1)\phi(x_2) [\rho+ \rho^2x_1x_2/2 + \rho^3 (x_1^2-1)(x_2^2-1)/6 +\rho^4 (x_1^3-3x_1)(x_2^3-3x_2)/24

+\rho^5 (x_1^4-6x_1^2+3)(x_2^4-6x_2^2+3)/120+\cdots ]

A good approximation is obtained truncating the series at \rho^3 term for |\rho| \le 0.4, and at \rho^5 term for 0.4 < |\rho|\le 0.7. Higher order terms may be required for |\rho| > 0.7.

Value

An approximation of bivariate normal cumulative distribution function.

References

Johnson, N. L. and Kotz, S. (1972) Continuous Multivariate Distributions. Wiley, New York.

Pearson, K. (1901) Mathematical contributions to the theory of evolution-VII. On the correlation of characters not quantitatively measureable. Philosophical Transactions of the Royal Society of London, Series A, 195, 1–47.

See Also

scoreCov

weightedScores documentation built on Aug. 23, 2026, 5:11 p.m.