Penalized unimodal spline density estimation with application to M-estimation

Classical M-estimation of a center mu from noisy data involves choosing a function rho then finding mu to minimize the sum of the values of the rho function evaluated at the residuals. The rho 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 rho function and provide inference. Convergence rates are given for the density estimator, and root-n convergence for the estimate (mu) over cap is attained without assumptions about the moments of the error density. (c) 2022 Elsevier B.V. All rights reserved.