Doctor of Philosophy
Statistics and Actuarial Sciences
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.
Summary for Lay Audience
We usually use probability to quantify our statement on the uncertain events in our daily life. However, sometimes we should not ignore the uncertainty in our probability assessment. Let us play a simple game. If we look at the left part of Figure 0.1 (on Page iii of the thesis), at your first glance, does it look like a rabbit or duck? Then turn to the right part to answer the same question. You may find that your answer on “whether it is more like a rabbit or duck” is different for these two parts, but actually, they are the same except for the difference in rotation angle. Under various viewing angles, our probability assessment of this image could be different.
We can see that sometimes there indeed exists uncertainty in the probability itself. In statistics, it is formally called model uncertainty. In this thesis, we have developed a new framework called the semi-G-structure to better quantify this kind of uncertainty from both a theoretical and practical side. Our vision is that this structure will bring a new theoretical perspective to the study of model uncertainty.
Li, Yifan, "Statistical Roles of the G-expectation Framework in Model Uncertainty: the Semi-G-structure as a Stepping Stone" (2022). Electronic Thesis and Dissertation Repository. 8928.
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