The Repeated Median Intercept Estimator: Influence Function and Asymptotic Normality

Given the simple linear regression model Yi = α + βXi + ei for i = 1, ..., n, we consider the repeated median estimator of the intercept α, defined as α̂n = medi medj,j ≠ i (XjYi − XiYj)/(Xj − Xi). We determine the influence function and prove asymptotic normality for α̂n when the carriers Xi and error terms ei are random. The resulting influence function is bounded, and is the same as if the intercept is estimated by the median of the residuals from a preliminary slope estimator. With bivariate gaussian data the efficiency becomes 2/π ≈ 63.7%. The asymptotic results are compared with sensitivity functions and finite-sample efficiencies.