Optimal Bias Robust Regression Psi and Rho Revisited

Abstract This paper is focused on detailed aspects of the loss function rho and its derivative psi for an optimal bias robust regression method that minimizes the maximum asymptotic bias subject to a constraint on normal distribution efficiency. The analytic form of the psi function was discovered by Yohai and Zamar (1997), but the analytic form of the rho function was not known. Furthermore, the psi function has a curious feature of being equal to zero on an interval around the origin which forms a psi function “flat spot” whose length decreases with increasing normal distribution regression estimator efficiency, and is hardly noticeable in a plot of the psi function for 95% normal distribution efficiency. This paper focuses on the following aspects the optimal bias robust regression estimator: (1) The analytic form for the rho function; (2) A computational problem posed by the psi function flat spot and a solution to the problem using a slightly modified psi function that entails little loss of bias robustness; (3) A suite of functions available in an R package for computing the optimal psi and rho functions, and the modified psi and rho functions, for a specified normal distribution efficiency