I am collecting $i=1\ldots N$ measurements of $x_i,y_i$ and computing the average ratio: $r=\frac{1}{N}\sum_{i=1}^N \frac{x_i}{y_i}$. The mean ratio is supposed to be close to 0.5, if that matters and $y_i > 0$.
However, I have errors in $y$, so that in fact $y_i-\epsilon_i$ ($\epsilon_i$ is unmeasured and also $y_i-\epsilon_i > 0$). I can put an upper bound on the error, such that $|\epsilon_i| \leq \varepsilon$. I want to obtain some confidence bounds on my mean estimate of $r$. Could come as $\pm$ interval or if there exists some form of a probability distribution that captures this (?). I don't know the distribution of error $\epsilon_i\sim\mathcal{P}?$
$\hat{r}-r=\frac{1}{N}\sum_{i=1}^N \frac{x_i}{y_i}-\frac{1}{N}\sum_{i=1}^N \frac{x_i}{y_i-\epsilon_i} = \pm?(\varepsilon)$ or $\sim\mathcal{P}(\varepsilon)?$