The PI deals with the bootstrap inferential strategies for two broad classes of bootstrap methods in the context of semi-nonparametric models. As a general-purpose approach to statistical inferences, the bootstrap has found wide applications in semi-nonparametric models. Unfortunately, systematic theoretical studies on the bootstrap inferences are extremely limited, especially when the nonparametric component is not root-n estimable. Two classes of bootstrap methods are considered: the exchangeably weighted bootstrap (EWB) and the model-based bootstrap (also known as the parametric bootstrap). The PI proves that the EWB consistently estimates the asymptotic variance of the Euclidean estimate and is theoretically valid in drawing semiparametric inferences in the framework of penalized M-estimation. However, the EWB may become invalid in drawing inferences for nonparametric components. Hence, the PI considers the model-based bootstrap, and theoretically justifies it as an universally valid inference procedure for all the parameters in semi-nonparametric models. The proposed research also involves the development of advanced empirical processes tools. The above research lays the theoretical foundation for the general semi-nonparametric inferences via various bootstrap sampling schemes, and establishes a general framework for non-standard asymptotic theory concerning the nonparametric components.
The immediate need for fast and efficiently extracting information from all the dimensions of modern massive data sets gives rise to the increasing popularity of the semi-nonparametric models. For example, to understand the recent financial crisis, the semi-nonparametric copula models are applied to address tail dependence among shocks to different financial series and also to recover the shapes of the impact curve for individual financial series. The proposed research promotes the use of semi-nonparametric models in analyzing modern complex data by developing a series of innovative and valid bootstrap inferential tools, and eventually gain substantial scientific productivity across various disciplines. Statistical science benefits from the increasing number of researchers trained in semi-nonparametric modelling both from the statistical and scientific viewpoints. This would include the students funded by this work, broader collaborating research and educational activities. The above research also produces easy-to-implement software for the public.
