Abstract
In the simple linear regression problem $\{Y_i = \alpha + \beta x_i + e_i i = 1,\cdots, n, e_i$ i.i.d. $\sim F$ continuous, $x_1 \leqq \cdots \leqq x_n$ known, $\alpha, \beta$ unknown$\}$ we investigate the following type of estimator: To each $s_{ij} = (Y_j - Y_i)/(x_j - x_i)$ with $x_i < x_j$ attach weight $w_{ij}$ and as estimator for $\beta$ consider the median of this weight distribution over the $s_{ij}$. A confidence interval for $\beta$ is found by taking certain quantiles of this weight distribution. The asymptotic behavior of both is investigated and conditions for optimal weights are given.
Citation
Friedrich-Wilhelm Scholz. "Weighted Median Regression Estimates." Ann. Statist. 6 (3) 603 - 609, May, 1978. https://doi.org/10.1214/aos/1176344204
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