The problem is: I have bivariate, normally distributed observations (x,y) to a model. But as I can't observe the original variables, I will observe them as (x+p,y+q) where p and q are some error terms which are assumed to be normal. Will the new data follow bivariate normal, or something else?
The new data will be bivariate normal, but: you can expect attenuated correlations and a decrease in power. This is because, since you have to evaluate A = X + p and B = Y + q, there are more sources of variation. E[A] = E[X + p] = E[X] + E[p], assuming that p is independent of X. Likewise, E[B] = E[Y + q] = E[Y] + E[q]. So your means for X and for Y will shift by mean(p) and mean(q), respectively. That's not so bad. But the increased variance comes about thusly: Var(A) = Var(X + p) = Var(X) + Var(p), again assuming p is independent of X, and Var(B) =Var(Y + q) = Var(Y) + Var(q), Var(A,B) = Var(A) + Var(B) + Cov(A,B) = Var(X) + Var(Y) + Cov(X,Y) + Var(p) + Var(q).
Well done Mary, but I think you might be doing BHARGAVARAMA's math stat homework!!
Mary and John, Thanks for your concern. I did this work already and this is not my home work :)- But my doubt is How we can conclude that (A,B) will be distributed as Bivariate normal even though A and B are normal variables.? There is a condition that to be checked that any linear combination of A and B is again a normal. This will be useful for me in my research doing on Measurement Error Models...
Once again thanks for your answers....
I was just joking around ..... I think you just need the theorem that any linear combination of multivariate normal random variables is again multivariate normal, There is not need to check conditions because this is a theorem which characterizes multivariate normal distributions.
See the link: http://www.statlect.com/normal_distribution_linear_combinations.htm
Hi john sir, I understood you were joking. We know that the linear combination of INDEPENDENT normal random variables is normal.But in the case of regression x and y need not to be independent. Thats where i am confusing...
Thanks in advance sir....
Bhargavarama, since (X,Y) as bivariate normal, then f(X,Y) is a function of g(X), h(Y), and their covariance. If the marginal distributions g(X) and h(Y) are normal, then you want to know if g(X+p) and h(Y+q) are normal, since the latter are marginal distributions of (A,B). I'll end this with every mathematician's favorite phrase "It can be shown..."
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