Robust Semiparametric Time-Delay and Doppler Estimation: Analysis of $R$ - and $M$-Estimators

This paper investigates time-delay and Doppler estimation in the presence of unknown heavy-tailed disturbances. Traditional approaches, such as the maximum likelihood estimator, achieve optimal mean squared error performance under the unrealistic assumption of perfect prior knowledge of the noise distribution. To address this limitation, previous work introduced a rank-based and distribution-free <tex>$R$</tex>-estimator, which is shown to be parametrically efficient, attaining the classical Cramér-Rao Bound irrespective of the unknown noise distribution, provided it belongs to the family of Complex Elliptically Symmetric distributions. The aim of this paper is to analyse and compare the performance of the <tex>$R$</tex>-estimator with an <tex>$M$</tex>-estimator, a widely used robust estimation approach. Specifically, we analyse their statistical efficiency for the time-delay and Doppler estimation problem, under various noise conditions. Furthermore, we propose to combine both estimators, leveraging their complementary strengths to enhance estimation performance. Numerical simulations illustrate the benefits of this hybrid approach.