Approximation of bivariate standard normal cumulative distribution function (Johnson and Kotz, 1972).
approxbvncdf(r,x1,x2,x1s,x2s,x1c,x2c,x1f,x2f,t1,t2)The correlation parameter of bivariate standard normal distribution.
\(x_1,\) see details.
\(x_2,\) see details.
\(x_1^2.\)
\(x_2^2.\)
\(x_1^3.\)
\(x_2^3.\)
\(x_1^4.\)
\(x_2^4.\)
\(\Phi(x_1)\Phi(x_2),\) where \(\Phi(\cdot)\) is the cdf of univariate standard normal distribution.
\(\phi(x_1)\phi(x_2),\) where \(\phi(\cdot)\) is the density of univariate stamdard normal distribution.
An approximation of bivariate normal cumulative distribution function.
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 \(|\rho| > 0.7\).
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.