Weighted Median Regression Estimates

Abstract

In the simple linear regression problem {Yi = 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 $x_i < x_j$ 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.