Towards a Theoretical Foundation of the Model-free Bootstrap for Regression and Time Series

The model-free bootstrap (MFB), first introduced in Politis [2013] followed by the monograph of Politis [2015], and further investigated in a series of papers (cf. Pan and Politis [2016b], Chen and Politis [2019], Das and Politis [2020], etc), is a recent advent in the bootstrap literature. The principle of MFB is to (invertibly) transform the original data to a space of i.i.d. variables, wherein the standard i.i.d. bootstrap is performed for the variables, and then the inverse transform is used to obtain bootstrap samples in the original data space. Because of the wide selection of applicable transforms, the MFB framework can be easily extended for complex data scenarios – such as regression and time series. The term "model-free" relates to the fact that thetransformations are estimated without model assumptions, i.e., nonparametrically.The main purpose of this dissertation is to build a theoretical foundation for the MFB under different setups. Specifically, in Chapter 1, we analyze the MFB under model-free regres- sion setup, and compare it with other methods of interest focusing on conditional coverage of prediction intervals. The concept of pertinent prediction intervals is extended to this setup, and we propose the notion of conjecture testing for predictive inference. In Chapter 2, we establish bootstrap validity of various statistics for a general class of univariate time series under very basic dependence assumptions, using the recently developed m ́approximation technique. In Chapter 3, the MFB algorithm of chapter 2 is further extended to handle multivariate time series, wherein we specifically focus on the generation of prediction regions. We also link the MFB under multivariate context with the well-known copula models. Finally in Chapter 4, we propose a new MFB algorithm for quantile autoregressive processes based on the MFB framework for general Markov process by Pan and Politis [2016b], using an augmented quantile regression estimator