This thesis also develops methodologies for carrying out inference using the semi-parametric DRM in the presence of distributed data. Techniques were developed for carrying out the dual empirical likelihood ratio test, which allows for the testing of composite hypothesis about DRM model parameters. This thesis also develops theories for the estimation of the baseline and marginal distribution functions for each sample alongside providing a method for estimating the quantiles of each distribution function. Applying the ADMM algorithm in the presence of distributed data, we have successfully fit a DRM and carried out inference which arrive to the same statistical conclusions as if the model was fit and inference was carried out using the same data in a non-distributed setting.