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.
Value
An approximation of bivariate normal cumulative distribution function.
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.
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.