The composite autoregressive system can be used to estimate a speech source-filter decomposition in a rigorous manner, thus having potential use in glottal inverse filtering. By introducing a suitable prior, spectral tilt can be introduced into the source component estimation to better correspond to human voice production. However, the current expectation-maximisation based composite autoregressive model optimisation leaves room for improvement in terms of speed. Inspired by majorisation-minimisation techniques used for nonnegative matrix factorisation, this work derives new update rules for the model, resulting in faster convergence compared to the original approach. Additionally, we present a new glottal inverse filtering method based on the composite autoregressive system and compare it with inverse filtering methods currently used in glottal excitation modelling for parametric speech synthesis. These initial results show that the proposed method performs comparatively well, sometimes outperforming the reference methods.