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Estimating the offset parameters of a mixture in the Fourier domain

In this contribution we present an algorithm for estimating some parameters of offset in the case of incomplete data. This estimation cannot be performed directly with an EM or SEM method because the density of local extrema in the likelihood map grows exponentially with the number of observations and because the SEM method provides a monotonic sequence of estimates so that bad initialization cannot be recovered. We perform the estimation in the Fourier domain. The offsets in time domain are transformed into pulsations in the Fourier domain. We minimize a quadratic distance between the parametric and empirical sampled Fourier transform with an EM method. Contrary to the problems encountered in the time domain the asymptotic loglikelihood of the sampled empirical Fourier transform is continuous w.r.t. the parameters of offset. We discuss the influence of the frequencies at which the Fourier transform is sampled and we present a numerical study of convergence of the proposed algorithms.