In the simple linear regression problem ᵢ = α + β xᵢ + eᵢ i = 1,⋯, n, eᵢ i.i.d. ~ F continuous, x₁ ⋯ xₙ known, α, β unknown\ we investigate the following type of estimator: To each sᵢⱼ = (Yⱼ - Yᵢ)/(xⱼ - xᵢ) with xᵢ < xⱼ attach weight wᵢⱼ and as estimator for β consider the median of this weight distribution over the sᵢⱼ. A confidence interval for β is found by taking certain quantiles of this weight distribution. The asymptotic behavior of both is investigated and conditions for optimal weights are given.
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Friedrich-Wilhelm Scholz (1978) studied this question.
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