Inconsistency in different ways of calculating CDF of bivariate

*To*: mathgroup at smc.vnet.net*Subject*: [mg106405] Inconsistency in different ways of calculating CDF of bivariate*From*: Stephen <aultman.stephen at gmail.com>*Date*: Mon, 11 Jan 2010 18:53:07 -0500 (EST)

I would like to calculate the CDF of a bivariate normal in some C++ code. My first thought was to reduce the dimensionality of the integral by using the conditonal distribution of the bivariate normal when one variable is known (see link below). When I try to calculate the CDF this way I get an incorrect result. In the code pasted below I defined a function, temp1, that uses mathematica's built in bivaraite normal CDF function. The function temp2 attempts to calculate the same quantity using a one dimension integral and the conditional distribution of the bivariate normal. It is more than just rounding errors. For some values of rho, the error is as much as 0.03, which is a lot in probability terms. Mostly I am curious why this doesn't work from a mathematical statistics perspective. There are ways to code around this, but I am curious if there is a bigger math issue at play. Any thoughts? mu1 = 0; mu2 = 0; sigma1 = 1; sigma2 = 1; rho = .1; z1 = 0; z2 = 0; Needs["MultivariateStatistics`"] temp1[mu1_, mu2_, sigma1_, sigma2_, rho_, z1_, z2_] := CDF[MultinormalDistribution[{mu1, mu2}, {{sigma1^2, rho*sigma1*sigma2}, {rho*sigma1*sigma2, sigma2^2}}], {z1, z2}] temp2[mu1_, mu2_, sigma1_, sigma2_, rho_, z1_, z2_] := NIntegrate[ PDF[NormalDistribution[mu1, sigma1], x]* CDF[NormalDistribution[ mu2 + (sigma2/sigma1)*rho (x - mu1), (1 - rho^2)*sigma2^2], z2], {x, -Infinity, z1}] Plot[temp1[mu1, mu2, sigma1, sigma2, rho, z1, z2] - temp2[mu1, mu2, sigma1, sigma2, rho, z1, z2], {rho, -.99, .99}] http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Conditional_distributions