Abstract
Financial risk is objective in modern financial activity. Management and measurement of the financial risks have become key abilities for financial institutions in competition and also make the major content in finance engineering and modern financial theories. It is important and necessary to model and forecast financial risk. We know that nonlinear expectation, including sublinear expectation as its special case, is a new and original framework of probability theory and has potential applications in some scientific fields, specially in finance risk measure and management. Under the nonlinear expectation framework, however, the related statistical models and statistical inferences have not yet been well established. In this paper, a sublinear expectation nonlinear regression is defined, and its identifiability is obtained. Several parameter estimations and model predictions are suggested, and the asymptotic normality of the estimation and the mini-max property of the prediction are obtained. Finally, simulation study and real data analysis are carried out to illustrate the new model and methods. In this paper, the notions and methodological developments are nonclassical and original, and the proposed modeling and inference methods establish the foundations for nonlinear expectation statistics.
Highlights
Finance is the core of economy, and financial safety is directly related to economic safety
It is well known that among all the assumption conditions imposed to the classical statistical models, the most vital one is that the models under study have a certain probability distribution that may or may not be known
The classical linear expectation and determinant statistics are built on such a distribution certainty or model certainty
Summary
Finance is the core of economy, and financial safety is directly related to economic safety. Under the classical statistics frameworks, including parameter models, nonparametric models, Bayes models, and time series models, the defined expectations are of linearity. Without this linearity, it is essentially difficult or impossible by the classical methods to achieve the classical certain conclusions, such as estimation consistency, asymptotic normality of the estimation. The new model tends to use a large value to predict response variable and obtains the mini-max prediction risk It implies that our method is a robust strategy and has potential applications in finance risk measure and management. The proofs of the theorems and the definition of the sublinear expectation space are postponed to appendices
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