On 2/10/2008, at 11:02 AM, Ravi Varadhan wrote:
I think it is meaningful to ask for a non-trivial Pr (X < x, Y=y)
when you
are writing down the likelihood for parameter estimation. This is
commonly
the case in likelihood estimation in bivariate failure time
models. If one
interprets Pr(Y=y) as the density evaluated y then:
Pr(X<x,Y=y) = Pr(X<x | Y=y) * f(y)
In R:
Pr(X<x,Y=y) = pnorm(x, mu=mu[1] + Sigma[1,2]*(y-mu[2])/Sigma[2,2],
sd=sqrt(Sigma[1,1] - (Sigma[1,2]^2)/Sigma[2,2])) * dnorm(y, mu=mu[2],
sd=sqrt(sigma[2,2]))
Fair enough; don't know if this was what Sasha is after, but I guess it
could be.
One should be careful with one's terminology however. Loose lips
sink ships.
Likelihoods are not in general probabilities which, as I am given to
understand, is why Fisher coined the usage ``likelihood'' in this
context.
cheers,
Rolf
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