## 1 Introduction
### Objective of the Study  
According to the above literature overview, previous work on SS estimation assumes that the FED are normal i.i.d In order to assess violations of this assumption, there are many solution methods in the recent literature, such as considering the FED as a gamma distribution or log-normal distribution, using the GARCH model to consider the Heteroskedasticity phenomena, and using EVT to take into consideration the occurrence of extreme demands. However, the performance of these methods is not guaranteed because there is an absence of comparative studies. Consequently, the final objective of this research can be summarized by the following primary research question. How can we improve the SS estimation methods to simultaneously consider the Heteroskedasticity phenomena and the occurrence of extreme demands?  
This work proposes to estimate SS using two methods mentioned above, which are also largely applied in finance and insurance domains to estimate Value at Risk which is defined as the quantile of the studied distribution for a given confidence level. The first method consists of combining simulation and GARCH methods, named filtered historical simulation (FHS), while the second one consists of combining EVT with the GARCH model, named conditional extreme value theory (CEVT). The choice to combine EVT and simulation with GARCH can be justified also by the success of these methods in providing a sufficient estimation of VaR in the risk measurement context. For each method, in the first step GARCH parameters we extract standardized residuals, and then, in the second step, we apply either EVT or historical simulation to residual series obtained from the GARCH model. This two-step procedure is different from traditional combination techniques, such as that applied by [PERSON] et al. [22] to combine kernel estimation and GARCH methods. To the best of our knowledge, this is the first work to evaluate SS using these two methodsand that compares their performance against traditional methods assuming a demand that follows the AR process with nonnormal distributions for residuals. We follow relatively similar methodology as [PERSON] et al. [22] with some divergence, especially in the case study and in the choice of distribution parameters. [PERSON] et al. [22] used an optimization framework to determine the optimal weight of each method used in combination. On the contrary, in our paper, the combination is based on each of the simulation and EVT methods applied to standardized residuals obtained from the GARCH model. Furthermore, the combination of GARCH models are very useful in the estimation methods for other areas, such as stock price volatility estimation [31], value at risk estimation [32], crude oil price volatility estimation [33], etc.  
The novelty of this research is about the proposition of new combined methods for SS estimation in the case when forecast errors are not normal i.i.d and to simultaneously consider the Heteroskedasticity phenomena and the occurrence of extreme demands. To the best of the authors' knowledge, this is the first time a quantile combination using FHS and CEVT has been used to calculate the SS. To validate the superiority of the proposed methods, both simulation and real data from a case study are applied. In the simulation, three cases are studied as follows. Case 1: when the forecast error follows a normal distribution, Case 2: when the forecasting error follows a lognormal distribution, and Case 3: when the forecasting error follows a gamma distribution. Moreover, to validate the statistical significance of the obtained simulation results, the one-way ANOVA technique is applied.