Estimate ratio of two expectations by sample means

Question

I have a question about the estimation of a ratio of two expectations. Suppose $X_{i}$ and $Y_{i}$ are two random variables with $i=1,\cdots,N$. We seek to estimate $\mathbb{E}X_{i}/\mathbb{E}Y_{i}$ by replacing $\mathbb{E}\left(\cdot\right) $ with the sample mean. This will introduce a bias, as $$ \mathbb{E}\frac{\frac{1}{N}\sum_{i}X_{i}}{\frac{1}{N}\sum_{i}Y_{i}}\neq \frac{\mathbb{E}X_{i}}{\mathbb{E}Y_{i}}\text{.}% $$ I was wondering if this can be solved, to some required (stochastic) order, by some analytical method (i.e. not jackknife or bootstrap).