R help
Rolf Turner <rolf at math.unb.ca> writes:
Peter Dalgaard writes:
Shutnik <shutnik_xx at yahoo.co.uk> writes:
Hello, I have the normal random variables y(t)~N(mu, sigma.sq) and want to decompose them into n normal variables: y(t) = x(t,1) +
+ x(t,n)
I presume this means y(t) = x(t,1) + ... + x(t,n) (R.T.)
x(t,i)~N(mu, sigma.sq/n)
I presume you want x(t,i)~N(mu/n, sigma.sq/n), elsewise the question doesn't make sense. I also presume you want the x(t,i) to be independent, elsewise the question is trivial. (R.T.)
The problem is not as simple as can appear. All my experiments didnt give me anything so far. Are there any tools to do this?
This should work, provided I understand the problem correctly: x <- rnorm(n,sd=sqrt(sigma.sq/n)) x <- x - mean(x) + y/n
I don't think it's that simple: By my calculations, Var(x_i) = 2*sigma.sq/n - sigma.sq/n^2, not sigma.sq/n.
Try again... Var(y/n) = sigma.sq/n^2, not sigma.sq/n so it cancels the second term rather than doubling the first.
I think the problem is actually fairly subtle (although I may be overlooking something simple). Something like a Gramm-Schmidt approach should work, but I can't quite suss it out.
You just need to orhtogonalize against the vector (1,1,...,1), which is what I did, effectively. The residuals x-mean(x) are independent of mean(x) by the Fisher-Cochran theorem (if I remember the name correctly...) and y/n has same distribution as mean(x) so you can substitute y/n for mean(x) and paste things back together again. But where does the t in y(t) and x(t,i) enter in all of this??
O__ ---- Peter Dalgaard Blegdamsvej 3 c/ /'_ --- Dept. of Biostatistics 2200 Cph. N (*) \(*) -- University of Copenhagen Denmark Ph: (+45) 35327918 ~~~~~~~~~~ - (p.dalgaard at biostat.ku.dk) FAX: (+45) 35327907