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st: RE: integrals - 2nd correction!


From   "FEIVESON, ALAN H. (AL) (JSC-SD) (NASA)" <alan.h.feiveson@nasa.gov>
To   "'statalist@hsphsun2.harvard.edu'" <statalist@hsphsun2.harvard.edu>
Subject   st: RE: integrals - 2nd correction!
Date   Fri, 22 Nov 2002 14:54:12 -0600

Dear Statalist and Clara - I must apologize again - I was in too much of a
hurry. I'll try this again. The bottom line is that I believe you can
evaluate this integral in terms of the 1-dimensional cumulative normal PHI
(-norm( ) - in Stata) and the 1-dimensional normal density fhi (-normd- in
Stata). It's messy , though. Here's an outline of how to proceed.

1. Suppose X and Y are random variables whose joint density is f(x,y)
(bivariate normal). Let E(X)=Ux, E(Y)=Uy , Var(X)=Sxx, Cov(X, Y)=Sxy and
V(Y)=Syy.

2. The joint density can be factored : f(x,y) = g(y|x)*h(x) where g(y|x), as
a function of y for fixed x, is the density of the conditional distribution
of Y given X=x, and h(x) is the marginal density of X.

3. Let I(t1,t2)F(t)dt denote the integral of a function F(t) with respect to
t, from t1 to t2. Let a = 0.56.
   
Then  I(-oo, a)f(x,y)dy =h(x)I(-oo,a)g(y|x)  = h(x)*P(Y<a|X=x).

But it is known (see Anderson, for example) that the distribution of Y given
X=x, is normal with mean Uy+B*(x-Ux) and variance Syy-B*B/Sxx, where B =
Sxy/Sxx.

Thus P(Y<a|X=x) = PHI[     (a - Uy-B*(x-Ux)  ) / sqrt(Syy-B*B/Sxx)   ] =
PHI(c*x + d), for known constants c and d.

Thus I(-oo, a)f(x,y)dy =h(x)*PHI(c*x + d), where h(x) = (1/sqrt(Sxx)*fhi(
(x-Ux)/sqrt(Sxx)). 

4. Then your desired integral is the integral of x*h(x)*PHI(c*x + d) with
respect to x. Recall h(x) is the marginal density of X, so h(x) =
(1/sigx)*fhi( (x-Ux)/sigx ) , where fhi(z)=(1/sqrt(2*_pi) )*exp(-z*z/2) and
sigx=sqrt(Sxx).

5. Therefore, your desired integral is of the form

I = I(-oo, x) x*fhi(k*x + h)*PHI(c*x + d), where k, h, c and d are known
functions of the means and variances/covariances.

6. Now this integral can be obtained using integration by parts.
Differentiate PHI(c*x + d )  and integrate x*fhi(k*x + h ). The latter can
be obtained using the fact that d/dz fhi(z) = -z*fhi(z). (Messy details left
to the user!)

7. If you don't feel like doing Step 6 in closed form, you can always
evaluate it with a 1-dimensional numerical integration.

8. Good luck!

Al Feiveson



-----Original Message-----
From: Clara [mailto:bldvegac@xa.bs.ehu.es]
Sent: Friday, November 22, 2002 10:26 AM
To: statalist@hsphsun2.harvard.edu
Subject: st: integrals


hi, I need to solve a double integral. This integral does not have an
analytic solution, then i need to do this numerically. There are someone
that know how can i do this in stata??

I need to integral:
 x*f(x,y)dxdy

where f(x,y) is a bivariate normal density function.
the integration intervals are: [-infinity, 0.56]

thanks, clara   



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