Statistical Roles of the G-expectation Framework in Model Uncertainty: the Semi-G-structure as a Stepping Stone
Abstract
The G-expectation framework is a generalization of the classical probability system based on the sublinear expectation to deal with phenomena that cannot be described by a single probabilistic model. These phenomena are closely related to the long-existing concern about model uncertainty in statistics. However, the distributions and independence in the G-framework are quite different from the classical setup. These distinctions bring difficulty when applying the idea of this framework to general statistical practice. Therefore, a fundamental and unavoidable problem is how to better understand G-version concepts from a statistical perspective.
To explore this problem, this thesis establishes a new substructure called the semi-G-structure. The semi-G-structure plays a hybrid role connecting the classical and G-framework. The semi-G-independence preserves the symmetry (which can depict both spatial and temporal situations of model uncertainty) and it is also related to the G-independence (which only describes temporal situations due to asymmetry). We prove a semi-G-CLT generalizing the classical CLT under model uncertainty with symmetric independence. It reveals the central role of semi-G-normal in the semi-G-structure. To the extent of our knowledge, the semi-G-normal is the unique kind of normal in this picture allowing both the variance uncertainty and the connection between univariate and multivariate objects.
Building on the semi-G-structure, we construct a series of data experiments to show the statistical insights of G-version concepts. This is the first time a statistical experiment has been constructed to illustrate the asymmetry in the G-independence. We further develop a nonparametric test of model uncertainty. We also discover a tight connection between sublinear expectations and interval-valued data through the semi-G-structure. As a financial application, we discuss a robust portfolio optimization problem under general covariance uncertainty with a more delicate view of the variance uncertainty. Furthermore, the semi-G-structure can address statistical questions related to model uncertainty, which could not be formulated or addressed easily in either classical or G-framework. In short, the semi-G-structure reveals an intrinsic connection between the classical and G-framework. Such a connection is beneficial to the study of model uncertainty by providing a new theoretical perspective with statistical flexibility. This is the vision of the thesis.