Maximum L-q-Likelihood estimation in functional measurement error models

Abstract

We consider a robust parametric procedure for estimating the structural parameters in functional measurement error models. The methodology extends the maximum Lq-likelihood approach to the more general problem of independent, but not identically distributed observations and the presence of incidental parameters. The proposal replaces the incidental parameters in the Lq-likelihood with their estimates, which depend on the structural parameter. The resulting estimator, called the maximum Lq-likelihood estimator (MLqE) adapts according to the discrepancy between the data and the postulated model by tuning a single parameter q, with 0 < q < 1, that controls the trade-off between robustness and efficiency. The maximum likelihood estimator is obtained as a particular case when q = 1. We provide asymptotic properties of the MLqE under appropriate regularity conditions. Moreover, we describe the estimating algorithm based on a reweighting procedure, as well as a data-driven proposal for the choice of the tuning parameter q. The approach is illustrated and applied to the problem of estimating a bivariate linear normal relationship, including a small simulation study and an analysis of a real data set.