Semi-Parametric Inference with Density Ratio Model Fitted to Distributed Data using Alternating Direction Method of Multipliers
Alexander Imbrogno · 2020
With the sheer volume and complexity of modern day data sets, there has become a need for new techniques and methodologies to handle problems related to "big data".One such problem of interest arrises when data is distributed and stored in various locations.For reasons of confidentially, complexity or volume, we are unable to have access to the entire dataset in one centralized location.Alongside the presence of distributed data, we are also interested in carrying out inference using all of the data from each local storage unit.This thesis presents the application of the Alternating Direction Method of Multipliers (ADMM) algorithm to fit a semi-parametric Density Ratio Model (DRM) to a collection of distributed independent samples.The model parameters obtained using ADMM were found to be comparable to those fit from the same data present in a non-distributed setting.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.Any length of writing will not be able to quantify my appreciation for my mother.Growing up, she would always tell me "You can do anything you set your mind to".The attitudes and beliefs these words have cultivated for me have been an extraordinary tool for helping me through the trials and tribulations of my research and thesis.She has lifted me up when times were tough and I know I can always count on her when needed.Thank you for everything that you have done and continue to do.I would like to thank my colleagues/ friends, Josh Miller and Marc Lapointe for always offering their thoughts, perspectives, and suggestions on my research and various works.Last but certainly not least, I would like to thank my best friends Thomas and Randy.Thank you guys for always supporting me on