## 6 Discussion
Their study reported the HDPE geomembrane reaching nominal failure based on the SCR, with the geomembrane showing no signs of actual rupture in the field, despite extremely low SCR values being observed at certain analysis points. Although the points mentioned above are considered and the three-stage degradation model taken into account (Figure 6), the geomembrane still appears to be in the second stage of degradation (antioxidant-depleted), as there was no evidence of chemical degradation in terms of changes in its tensile strength properties. Considering this, non-parametric statistics were applied, utilizing the KDE to obtain a probability density function (PDF) that best represented the behavior of the geomembrane as a function of tensile strength under the given boundary conditions (Figure 9). With the application of non-parametric statistics, for each data point, a function was created centered on those data, and the PDF was estimated according to the sum of these cores divided by the total number of specimens [84; 85]. Thus, it was not necessary to check the fit of the data to a previously known distribution. Figure 8: Variation in SCR comparing the average values of the samples with 50% of the value presented in the GRI GM13 standard, with statistical errors shown. Parametric statistical methods are quite widespread in the literature, but these methods come with a caveat: the requirements of a set of assumptions about the data's underlying distribution. In other words, this means that knowledge of the PDF for the data under study is a prerequisite. However, proposing an incorrect PDF can introduce estimation errors and compromise the reliability of the statistical conclusions. In contrast, non-parametric statistical methods offer a more flexible alternative, free from the constraints of distributional assumptions. Among these, methods based on the KDE have gained prominence [86]. In practice, the KDE provides a straightforward way to structure datasets without imposing a specific parametric model, making it a valuable tool in data analysis [87]. In situations where the sample data present marked asymmetry effects or the presence of outlier values, among other unusual situations, using strategies independent of the previous proposition of a PDF becomes a powerful option. Using the PDF obtained via the KDE is a straightforward approach that results in a smoother and more continuous representation of the proper distribution of the sample data. These PDFs (Figure 9), shaped by the KDE, can reflect local patterns in the data and capture nuances and details that may be missed using more rigid methods. Simultaneously, the KDE helped smooth out noise present in the data and provided more precise insights into the overall behavior. It is evident that for this specific exposure scenario and this particular geomembrane, the PDF does not fit any known parametric statistical distribution. A new sampling experiment might yield distinct PDFs. From a purely theoretical standpoint, to thoroughly analyze the behavior of these PDFs, it would be necessary to repeat the experiment multiple times in multiple scenarios and examine whether any consistent patterns emerge. This approach allows for a comprehensive understanding of the variability and characteristics of the PDFs. Given the multimodal nature of these PDFs, identifying any discernible patterns is particularly challenging, as the complexity and variability make it difficult to visualize a consistent trend. Altering the exposure conditions, time, and/or the geomembranes will result in different PDFs, which means that it is not possible to generalize the PDFs shown in Figure 9.  
Figure 9: Tensile strength PDF for (**a**) SP1; (**b**) SP2; (**c**) SP3. These PDFs enabled the determination of the probability of nominal failure of the geomembrane (when the tensile strength is reduced to 50% of the minimum value specified by the GRI GM13 standard [30, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45]) under the given time and exposure conditions as a function of tensile strength.