We apply several classes of stochastic multidimensional models to statistical analysis of earthquake catalogues using likelihood methods. We investigate the importance of including different earthquake parameters in the model: epicentral coordinates, hypocentral depth, time limits for interearthquake interaction, and especially spatial distribution of earthquakes as well as spatial aftershock patterns. Results of this study combined with other investigations, suggest that most distributions controlling earthquake interaction have a fractal or scale-invariant form. Developed models are used for statistical analysis of several earthquake catalogues to evaluate parameters of earthquake occurrence. These parameters are shown to be similar for shallow earthquakes of different magnitude ranges and seismogenic regions, confirming self-similarity of the earthquake process. Whereas intermediate earthquakes seem to emulate the pattern of shallow earthquake occurrence, albeit at a much smaller aftershock rate, deep earthquakes differ significantly in their properties. Predictability of standard shallow earthquake catalogues has been analysed; we present evidence that for the best available catalogues the predictability is close to 10 bits per earthquake. Several synthetic earthquake catalogues have been created and processed through the likelihood inversion scheme. The results from likelihood analysis of these catalogues confirm our approach.