joint distributions of the pairs.

\[ratio\_estimation\]

[**Ratio of two means**]{} Consider a setting where independent pairs of random variables $(X_1, Y_1), \ldots, (X_n, Y_n)$ are observed, and suppose that interest is in the ratio of the mean of $Y_i$ to the mean of $X_i$, that is $\theta = \mu_Y / \mu_X$, with $\mu_X = \expect_G(X_i)$ and $\mu_Y = \expect_G(Y_i) \ne 0$ $(i = 1, \ldots, n)$.

Assuming that sampling is from an infinite population, one way of estimating $\theta$ without any further assumptions about the joint distribution of $(X_i, Y_i)$ is to set up an unbiased estimating equation of the form (\[estimating\_eqautions\]), with $\psi^i(\theta) = Y_i - \theta X_i$. Then, the $M$-estimator is $$\label{M_ratio}
    \hat\theta = \arg \mathop{\rm solve}_{\theta \in \Re} \left\{\sum_{i = 1}^n \psi^i(\theta) = 0 \right\} = \frac{s_Y}{s_X} \, ,$$ where $s_X = \sum_{i = 1}^n X_i$ and $s_Y = \sum_{i = 1}^n Y_i$. The estimator $\hat\theta$ is generally biased, as can be shown, for example, by an application of the Jensen inequality assuming that $X_i$ is independent of $Y_i$, and efforts have been made in reducing its bias; see, for example, @durbin:1959, for an early work in that direction using the jackknife.

For the estimating function for 