Variance of weighted average when weights have a distribution

Question

Given two random variables $S$ and $W$, I'm trying to determine the variance of the weighted average

$$Z = \frac{\sum_{i=1}^n s_i w_i}{\sum_{i=1}^n w_i}$$

(With the same $w_i$ in denominator as in the numerator)

It seems $E[Z] = E[S]$, but what would be an estimate of the variance?

If the denominator term wasn't there could simply use the Variance of a Product [link]

Answer 1

Use the law of total variance:

$var(Z) = var_w[E[Z|W]) + E_w[var(Z|W)]$

Let's calculate the two intermediate values first:

1. $$ E[Z|W] = \frac{\sum_i E[s_i]w_i}{\sum_i w_i} = E[S]$$ which does not depend on w, thus $var_w[E[Z|W]] = 0$

2.

if we assume independence of $s_i$'s

$$var(Z|W) = \frac{var(\sum_i w_i s_i)}{(\sum_i w_i)^2}$$

$$ = \frac{\sum_i w_i^2}{(\sum_i w_i)^2} var(S)$$

thus $E_w[var(Z|W)] = E[\frac{\sum_i w_i^2}{(\sum_i w_i)^2}]var(S)$

It seems the $E[\frac{\sum_i w_i^2}{(\sum_i w_i)^2}]$ part cannot be further simplified unless the distribution of $w_i$'s are given.

Answer 2

Without independence between $S$ and $W$, $E[Z] \ne E[S]$ in general. This is because $$E[Z|W]=\Sigma_i \frac{w_i}{\Sigma_j w_j}E[s_i|W].$$ Hence, without knowing the conditional distribution of $s_i$ given $W$, we don't know whether $E[s_i|W]$ is the same across all $i$.