Distributional uncertainty of the financial time series measured by $G$-expectation

Based on the law of large numbers and the central limit theorem under nonlinear expectation, we introduce a new method of using $G$-normal distribution to measure financial risks. Applying max-mean estimators and a small windows method, we establish autoregressive models to determine the parameters of $G$-normal distribution, i.e., the return, maximal, and minimal volatilities of the time series. Utilizing the value at risk (VaR) predictor model under $G$-normal distribution, we show that the $G$-VaR model gives an excellent performance in predicting the VaR for a benchmark dataset comparing to many well-known VaR predictors.