Recursive estimation in linear models with general errors and grouped data: a median-based procedure and related asymptotics

Abstract

We introduce in this paper an iterative estimation procedure based on conditional medians valid to fit linear models when, on the one hand, the distribution of errors, assumed to be known, may be general and, on the other, the dependent data stem from different sources and, consequently, may be either non-grouped or grouped with different classification criteria. The procedure requires us at each step to interpolate the grouped data and is similar to the EM algorithm with normal errors. The expectation step has been replaced by a median-based step which avoids doing awkward integration with general errors and, also, we have substituted for the maximisation step, a natural one which only coincides with it when the errors are normally distributed. With these modifications, we have proved that the iterative estimating algorithm converges to a point which is unique and non-dependent on the starting values. Finally, our final estimate, being a Huber type M-estimator, may enjoy good stochastic asymptotic proper-ties which have also been investigated in detail