st: calculation of bivariate normal probabilities

[Jenkins S P <stephenj@essex.ac.uk>]

Stata 9 provides the function -binormal(h,k,r)-, which returns the joint 
cumulative distribution of the bivariate normal with correlation r; 
cumulative over (-inf,h]x(-inf,k]. [The equivalent function in version 8.2 
was -binorm(h,k,r)-.)

I am not 100% confident about calculation of bivariate normal 
probabilities once we move outside the standard bivariate normal 
distribution (or indeed standard univariate normal distribution), and seek 
advice on this.

My understanding is that -binormal()- evaluates the double integral

        INT(-inf,h] INT(-inf,k) {  f(x1, x2)dx1dx2 }

where f(x1,x2) refers to the standard bivariate normal density with 
variates X1, X2 each having mean zero, unit variance, and correlation of 
r. (Put another way, the correlation matrix R has ones on the leading 
diagonal, and r in the off-diagonal cells)

By contrast, I want to evaluate

	INT(-inf,p] INT(-inf,q) { g(x1,x2)dx1 dx2 }

where g(x1,x2) refers to a general bivariate normal density with variates 
X1, X2 having means m1 and m2, variances v1, v2, and covariance c (the 
covariance matrix is C).

To use -binormal()- for the calculation, my inclination was to derive the 
correlation matrix R from covariance matrix C [-matrix R = corr(C)-] and 
to extract r from this, and also to set the upper integration points as 
follows:

	h = (p - m1)/sqrt(v1)
	q = (q - m2)/sqrt(v2)

Is this correct?

In fact, I am also seeking an answer for the general multivariate normal 
case. (If one had a function available to evaluate standard multivariate 
normal distribution probabilities analogous to -binormal()-, would 
evaluation for the general case proceed in a similar way? I.e. derive R 
from C, 'standardize' the upper integration points, and then substitute 
these into the function.)

Advice appreciated. (My web searches for information have provided closely 
related information but not the definite confirmation I seek.)

Stephen
=============================================
Professor Stephen P. Jenkins <stephenj@essex.ac.uk>
Institute for Social and Economic Research (ISER)
University of Essex, Colchester CO4 3SQ, UK
Phone: +44 1206 873374.  Fax: +44 1206 873151.
http://www.iser.essex.ac.uk

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