# Summation of median and quantiles of multiple forecasted variables

Assume that I have Y1_hat with its P10_1 and P90_1 and Y2_hat with its P10_2 and P90_2.

Is it valid to sum Y1_hat and Y2_hat, sum P10_1 and P10_2, and sum P90_1 and P90_2? and would that present any possible issues?

The problem I have is that I have to forecast revenue for specific "groups" but also, I have to forecast the total revenue which is represented by the sum of revenue coming from all groups. I cannot make two forecasting models where one is for "groups" revenue and the other is for total revenue because that will make a conflict in forecasting where if you sum the forecasting coming from groups, it will not equal the forecasting for total revenue. I also have to show quantiles for both cases.

I would appreciate a theoretical explanation that ensures non-crossing quantiles if valid, and validates that the sum of two or more predicted random variables distribution makes sense

• If $\hat Y_1$ and $\hat Y_2$ are expected values (means) then you can reliably sum them to give $\hat Y$. If not (e.g. medians or modes of the predicted distributions) then probably not. The quantiles cannot be reliably summed to give quantiles of the predicted sum. May 8 at 13:16
• I'd be really happy if you offered a rationale in the answers if you have the time @Henry And what would you do if you faced the issue I'm facing? May 8 at 16:25