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st: calculation of bivariate normal probabilities


From   Jenkins S P <stephenj@essex.ac.uk>
To   statalist@hsphsun2.harvard.edu
Subject   st: calculation of bivariate normal probabilities
Date   Wed, 25 May 2005 07:18:18 +0100 (BST)

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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