Percentile of sum of random variables

Solution 1:

In general, quantiles of sums of random variables are difficult. Even for small discrete samples useful results are difficult to find.

For example, if I know the sample means and sample sizes of two samples $x_i$ and $y_i$ of respective sizes $n,$ then I can find the mean of sample $z_i = x_i + y_i$ as $\bar z = (\sum_i x_i + \sum_i y_i)/n.$

However, I know of no similar relationship for the median. As a trivial example, the sum of $x = (1, 2, 3, 14, 15)$ and $y = (1, 2, 3, 14, 15)$ has vastly different quantiles than the sum of $x' = (1, 2, 3, 14, 15)$ and $y' = (15, 14, 3, 2, 1).$ [You might be able to get some useful results for sums of order statistics.]

Notes (1) The general intractability of combining quantiles is one reason the US Census Bureau is willing to release median incomes for census tracts, but (to protect individual privacy) not mean incomes.

(2) There are some perhaps tangentially related results. For example, if $X_i$ are i.i.d from a continuous population whose density function $f$ is positive in the vicinity of the median $\eta,$ then the sequence of sample medians $\tilde X_n$ is asymptotically normal with mean $\eta$ and asymptotic variance $[4nf(\eta)^2]^{-1}.$ And similar theorems apply for $p$th percentiles with $0 < p < 1$ (but not for maximums and minimums). [One reference is Bain & Englehardt (1992) p244.]