## 5 Conclusions  
The safety stock estimation needs forecast error uncertainty measurement. Deviation to the normal distribution and i.i.d assumptions lead to a problem in practice. Although certain forecasting models have analytical expressions for the variance over lead time, again retaining the same assumptions, these typically are not considered in practice. The use of empirical approaches, as developed in this research, overcomes these limitations, as the assumptions are relaxed and the error distribution considered originates from the final forecast, which can include any further adjustments or be based on any estimation method, irrespective of the existence of analytical expressions. The proposed combined estimation methods improve on the empirical approaches and achieve superior inventory performance. Decisively, this is conducted in a way that is easy to implement with existing forecasting systems, since the only input required is the historical forecast errors. A further advantage of the proposed approach is that is relevant to practice is that it is fully automatic and data-driven, and therefore, can be implemented in the context of supply chain forecasting. Empirical techniques can improve the computation of such SS. In particular, GARCH models face a heteroscedastic forecast error that varies over time. EVT takes into account the occurrence of extreme demands and historical simulation does not need to rely on a determined distribution. However, if forecast errors are heteroskedastic, do not follow a known distribution and there is a presence of extreme demand, then traditional approaches are unsuitable. Two combined empirical methods are proposed to determine the SS in a more robust fashion and compare them with traditional methods published in the literature, under different supply chain parameters. The first method is named FHS which combines the GARCH model with the historical simulation method, and the second method is named CEVT which combines GARCH with extreme value theory. To the best of our knowledge, any previous work uses one of these two combination approaches to calculate SS. Comparative analyses show the superiority of these combination methods with respect to the tick loss function for the different CSL targets and for shorter and longer lead times. In most cases, CEVT gives the lowest losses; otherwise, FHS takes its place followed by the GARCH method. The result remains in the case of simulated data with lognormal and gamma distributions and in the case of real data. Such results are confirmed using ANOVA. Figure 13: Main effects plot (data means) for TLF. That test is applied to TLF as an output response and Method, CSL, \(\phi\), LT, and FED are the input factors. The results of such tests indicate that the simple and proposed combined methods for SS calculation, the CSL, the autoregressive parameter and the lead time have an important influence on TLF and consequently in the decision taken on SS. In contrast, the forecast error distribution (FED) does not influence the decision. Regarding backorders and inventory investment, the results indicate that the GARCH-based methods outperform the traditional approaches for CSL targets. In particular, for real data and if the LT is short, GARCH-based methods always work better for most of the CSL except for the extreme level 99%, the GARCH achieves the best results. For longer LT, which remains except for the 99% level, the FHS attains the lowest backorders and inventory investment. According to [PERSON] and [PERSON] [53], little empirical work has been conducted in the context of combining conditional quantile forecasting, as in [PERSON] et al. [22] who presented the combination scheme based on two quantile forecasts, KDE and CGARCH. The proposed combination also improves the empirical approaches and achieves a superior inventory performance. Extending the current research on the other combined approaches based on the KDE and GARCH model is the first of our interesting perspectives.