paper

Semi--normal: a Hybrid between Normal and -normal (Full Version)

arXiv:2104.04910

Abstract

The -expectation framework is a generalization of the classical probabilistic system motivated by Knightian uncertainty, where the -normal plays a central role. However, from a statistical perspective, -normal distributions look quite different from the classical normal ones. For instance, its uncertainty is characterized by a set of distributions which covers not only classical normal with different variances, but additional distributions typically having non-zero skewness. The -moments of -normals are defined by a class of fully nonlinear PDEs called -heat equations. To understand -normal in a probabilistic and stochastic way that is more friendly to statisticians and practitioners, we introduce a substructure called semi--normal, which behaves like a hybrid between normal and -normal: it has variance uncertainty but zero-skewness. We will show that the non-zero skewness arises when we impose the -version sequential independence on the semi--normal. More importantly, we provide a series of representations of random vectors with semi--normal marginals under various types of independence. Each of these representations under a typical order of independence is closely related to a class of state-space volatility models with a common graphical structure. In short, semi--normal gives a (conceptual) transition from classical normal to -normal, allowing us a better understanding of the distributional uncertainty of -normal and the sequential independence.

arXiv comment109 pages, 8 figures, a comprehensive document for conference and open discussions, to be divided later for publications, readers may navigate to the parts they are interested in by the table of contents

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