Compensatory model for quantile estimation and application to VaR

In contrast to the usual procedure of estimating the distribution of a time series and then obtaining the quantile from the distribution, we develop a compensatory model to improve the quantile estimation under a given distribution estimation. A novel penalty term is introduced in the compensatory model. We prove that the penalty term can control the convergence error of the quantile estimation of a given time series, and obtain an adaptive adjusted quantile estimation. Simulation and empirical analysis indicate that the compensatory model can significantly improve the performance of the value at risk (VaR) under a given distribution estimation.