  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.
