The peculiar motions of early-type galaxies in two distant regions — VI. The maximum-likelihood Gaussian algorithm

The EFAR project is designed to measure the properties and peculiar motions of early-type galaxies in two distant regions. Here we describe the maximum-likelihood algorithm we developed to investigate the correlations between the parameters of the EFAR data base. One-, two- and three-dimensional Gaussian models are constructed to determine the mean value and intrinsic spread of the parameters, and the slopes and intrinsic parallel and orthogonal spread of the Mg2–Mgb′,Mg2–σ,Mgb′–σ relations, and the Fundamental Plane. In the latter case, the cluster peculiar velocities are also determined. We show that this method is superior to ‘canonical’ approaches of least-squares type, which give biased slopes and biased peculiar velocities. We test the algorithm with Monte Carlo simulations of mock EFAR catalogues, and derive the systematic and random errors on the estimated parameters. We find that random errors are always dominant. We estimate the influence of systematic errors resulting from the way clusters were selected, and the hard limits and uncertainties in the selection function parameters for the galaxies. We explore the influence of uniform distributions in the Fundamental Plane parameters and the errors. We conclude that the mean peculiar motions of the EFAR clusters can be determined reliably. In particular, the placement of the two EFAR sample regions relative to the Lauer & Postman dipole allows us to constrain strongly the amplitude of the bulk motion in this direction. We justify a posteriori the use of a Gaussian modelling for the galaxy distribution in the Fundamental Plane space, by showing that the mean likelihood of the EFAR sample is obtained in 10 to 30 per cent of our simulations. We derive the analytical solution for the maximum-likelihood Gaussian problem in N dimensions in the presence of small errors.