the ratio of means, $j(\theta) = s_X$, $u(\theta) = 0$, and, using (\[e\_mat\]) and (\[d\_mat\]), $e(\theta) = s_{YY} + \theta^2 s_{XX} - 2\theta s_{XY}$ and $d(\theta) = -s_{XY} + \theta s_{XX}$, where $s_{XX} = \sum_{i = 1}^n X_i^2$, $s_{YY} = \sum_{i = 1}^n Y_i^2$ and $s_{XY} = \sum_{i = 1}^n X_i Y_i$. So, the empirical bias-reducing adjustment in (\[general\_empirical\_adjustment\]) is $s_{XY}/s_{X} - \theta s_{XX}/s_{X}$, and the solution of the adjusted estimating equations results in the RB$M$-estimator $$\label{BR_ratio}
    \tilde\theta = \arg \mathop{\rm solve}_{\theta \in \Re} \left\{\sum_{i = 1}^n \psi^i(\theta) + A(\theta) = 0 \right\} = \frac{s_Y + \frac{s_{XY}}{s_{X}}}{s_X + \frac{s_{XX}}{s_{X}}} \, .$$ In this case, the empirical bias-reducing adjustment has the side-effect of producing an estimator that is more robust to small values of $s_X$ than the standard $M$-estimator is. In particular, as $s_X$ becomes smaller in absolute value, $\hat\theta$ diverges, while $\tilde{\theta}$ converges to $s_{XY}/s_{XX}$, which is the slope of the regression line through the origin of $y$ on $x$. As a result, when $\mu_X$ is small in absolute value, $\tilde\theta$ has not only smaller bias, as granted by the developments in the current paper, but also smaller variance than $\hat\theta$, and, hence, smaller mean squared error.

To illustrate the 