# minimax and transformations

for two real random $n$-vector $y$ and $x$ and a random $n$-vector $e$ with distribution $F$ independent of $x$ we know (1) that the estimator

$$\text{med}\left( \frac{y_i}{x_i}\right)$$

is minimax estimator for $\alpha$ in the model:

$$y_i=\alpha x_i+e_i$$

My question is the following: does this result implies that the estimator

$$\text{med}\left( y_i x_i\right)$$

is minimax estimator for $\alpha$ in the model

$$y_i=\alpha/x_i+e_i$$

(1) Martin,R.D. Yohai, V.J. and Zamar,R.H., Mini-max bias, robust regression. Ann. Stat. (1989)

Yes. Let $z_i = 1/x_i$. Use the initial result on the model $y_i=\alpha z_i + e_i$, for which all assumptions seem to be satisfied, and you get the result you want.