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 maximu m-likelihood algorithm we developed to investigate the correlations between the parameters of the EFAR data base. One-, two- and three-dimensional Gau ssian models are constructed to determine the mean value and intrinsic spre ad of the parameters, and the slopes and intrinsic parallel and orthogonal spread of the Mg-2-Mgb', Mg-2-sigma Mgb'-sigma relations, and the Fundament al Plane. In the latter case, the cluster peculiar velocities are also dete rmined. We show that this method is superior to 'canonical' approaches of l east-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 o f 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 Fundamen tal Plane parameters and the errors. We conclude that the mean peculiar mot ions of the EFAR clusters can be determined reliably. In particular, the pl acement of the two EFAR sample regions relative to the Lauer & Postman dipo le allows us to constrain strongly the amplitude of the built motion in thi s direction. We justify a posteriori the use of a Gaussian modelling for th e galaxy distribution in the Fundamental Plane space, by showing that the m ean likelihood of the EFAR sample is obtained in 10 to 30 per cent of our s imulations. We derive the analytical solution for the maximum-likelihood Ga ussian problem in N dimensions in the presence of small errors.