approxbvncdf: 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)

Value

An approximation of bivariate normal cumulative distribution function.

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 $|\rho| > 0.7$.

References

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