# Estimating component variance for a sum of random variables

Say I have two zero mean single variate independent random variables $X$ and $Y$, and a third variable $Z = X + Y$. I can draw samples $z_i$ with $i = 1..n$ from $Z$ and I know $Var(Y)$. How can I estimate $Var(X)$ based on the measurements $z_i$?

A trivial estimator of $\Sigma = Var(X)$ would be $\hat{\Sigma} = Var(z_i) - Var(Y)$. The problem is that $\hat{\Sigma}$ could actually be negative based on the drawn samples $z_i$. Is there an estimator (possibly biased) which always results in a valid variance?

I have a related question on the stats SE site.

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So take the samples of $z_i$ and calculate their variance. Subtract $Var(Y)$ from this to get your estimate of $Var(X)$.
As for your update, the only way $\hat \Sigma$ could be negative is if $Var(Z)$ is smaller than $Var(Y)$, and I'm pretty sure that can't happen if $Z$ is the sum of $X$ and $Y$ and those two variables really are independent. If $Var(z_i)$ is smaller than $Var(Y)$ because it is also smaller than $Var(Z)$ due to sampling errors, keep sampling until you get a valid variance.
Thanks. The problem is that lets say $\Sigma = Var(X)$ and the estimator $\hat{\Sigma} = Var(z_i) - Var(Y)$ as you suggested, $\hat{\Sigma}$ could actually be negative based on the drawn samples $z_i$. Not sure how to handle that. I'll update the question. – Jakob Feb 20 '13 at 15:36