0
$\begingroup$

Let $X$ be a random variable with variance $Var[X]=\sigma^2$. Given a number of (independent) realizations $X_i$, is it possible to estimate the variance of the inverse weighted sum: $ Var\left[\frac{1}{\sum_i a_iX_i}\right] $ ? What I am actually looking for is an estimate of the variance of the above expression without having a particular set of realizations - just knowing the weights $a_i$ and $\overline X$ and $\sigma^2$.

I found the delta method, which is essentially a kind of Taylor expansion: $ Var[f(\vec X)] \approx Var[X]\left(\sum_i\left(\frac{\partial}{\partial X_i}f(\vec X)\right)^2\right)\\ =Var[X]\left(\sum_i\left(-\frac{a_i}{2\left(\sum_i a_iX_i\right)^2}\right)^2\right) $ However, this expressions still has $X_i$ inside, which doesn't solve my problem.

  • 0
    @DilipSarwate: Thanks for this interesting comment! I did not think about this before. There might be a fundamental problem indeed.2012-09-20

0 Answers 0