Penalized unimodal spline density estimation with application to M-estimation

Abstract

Classical M-estimation of a center μ from noisy data involves choosing a function ρ then finding μ to minimize the sum of the values of the ρ function evaluated at the residuals. The ρ function can be optimized with some knowledge of the error density family, but this is usually unavailable a priori. We propose to estimate a truncated version of the error density, using a penalized spline density estimator that is constrained to be unimodal and symmetric, and to use the density estimate to determine the ρ function and provide inference. Convergence rates are given for the density estimator, and root-n convergence for the estimate μˆ is attained without assumptions about the moments of the error density.