0
$\begingroup$

Suppose I have 1000 draws each of two random variables X and Y.

If I wanted to sample the sum of these variables, I would simply calculate 1000 samples, i.e.

$$ S_{i}=X_{i}+Y_{i}, i=1,2,…,1000 $$

And that would give me draws from the pdf of the sum of these two variables.

Now suppose I know that the correlation between these two variables is ρ=0.8

A higher value of X is now more likely to coincide with a higher value of Y, and the standard deviation of the sums will be higher.

Can I resample the sum of the two variables without adding a specific assumption regarding the specific destribution of X and Y?

For example.

Assume X and Y are population forecasts for male population in Washington and Oregon in 2050. I have two separate models for population in Oregon and Washington that give me pdf for population in Washington/Oregon in 2050. Because the models were run independently of each other, X+Y=S will underestimate the standard deviance of the sum, simply because of the fact that coefficients and/or the future development of independent variables in the model for Washington and Oregon are likely to be correlated.

Let us also assume that I only have those draws and I cannot account for the fact that X and Y are related by designing a smarter model. I can however observe the correlation of population numbers in Washington and Oregon.

$\endgroup$
  • $\begingroup$ I don't understand the question. You have already described how to draw samples of the sum. $\endgroup$ – Tom Minka Oct 22 '14 at 11:13
  • $\begingroup$ Thank you for your reply. I know how to draw samples from the sum if I assume the two variables to be independent, they are however correlated. This fact is not reflected in my draws because I obtained these two sets of draws independently from each other. $\endgroup$ – mzuba Oct 22 '14 at 12:34
4
$\begingroup$

You cannot sample the sum without making distributional assumptions, but you can compute its variance, using the formula $$ {\rm var}(X+Y) = {\rm var}(X) + {\rm var}(Y) + 2 {\rm cov}(X,Y) $$ since ${\rm cov}(X,Y) = \rho \sqrt{{\rm var}(X) {\rm var}(Y)}$.

$\endgroup$
  • $\begingroup$ Thank you for your answer, although this is not what I had hoped for. It seems to me that I will have to make further assumptions. If I assume that Var1 and Var2 are normally distributed, I can compute the pdf of the sum. But what would the least restrictive distributional assumption be that would allow me to compute it? $\endgroup$ – mzuba Oct 29 '14 at 11:32
  • $\begingroup$ A copula model will allow you to purely model correlation between the variables whilst preserving the univariate distributions that you already have. $\endgroup$ – Tom Minka Oct 29 '14 at 14:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.