## 8 Extended Investigations on Method 2  
As mentioned in Section 4, the validity of method 1 had been reaffirmed in recent works (i.e., [24; 56]). Both in [27] and here, the initial validation of method 2 was performed by comparing--partially--the outcome of the two models discussed here. Further validation of method 2 can also be performed by comparing the outcome of the model parameters (together with their confidence interval) with typical gear data. Such validation is performed by examining the confidence intervals of \(k_{1}\) and \(k_{2}\). Confidence intervals are then compared with the typical values, which are the slopes of the limited-life and the long-life region for case-hardened materials according to ISO 6336-3 [6] and ANSI-AGMA 2001 [7]. The parameters \(N_{e}\) and \(\sigma_{e}\) were not considered. On the one hand, it is clear that in the database of Figure 9, the fatigue knee is not located at 3 million cycles (also see examples from Figures 10-14). On the other hand, discussion about \(\sigma_{e}\) cannot be conducted for two reasons. Firstly, different from ISO \(\sigma_{F\;lim}\) (i.e., the nominal bending stress number), \(\sigma_{e}\) cannot be seen as either an endurance limit or as a fatigue limit. Indeed, \(\sigma_{e}\) is just a parameter of Equation (5); therefore, it lacks all the statistical properties of an endurance (or fatigue) limit. Secondly, if one supposes that \(\sigma_{e}\) is equivalent to \(\sigma_{F\;lim}\) (or to ANSI-AGMA \(s_{at}\)), the comparison will become a comparison between the estimated material endurance limits and their standardized equivalent (which, in any case, is not the aim of this article). The methodology adopted here takes inspiration from previous works. One example is the work of [PERSON] et al. [74] in which they performed a Monte-Carlo simulation of a fatigue database composed of 188 test data in order to investigate how to properly estimate a SN curve. Then, several testing conditions were simulated and the convergency of the parameter confidence interval was compared, aiming to define the best sampling strategy. A similar procedure was adopted by [PERSON] and [PERSON] [75; 76], where 100 staircaseswere virtually simulated to validate their models by comparing the estimates. Similar methodologies for the evaluation of the behavior of stress-lifetime models can also be found in [53; 55; 77; 78]. Here, to increase the number of available experimental campaigns, as well as the numerosity of experimental campaigns with a high number of experimental points, some test series about similar gears made with comparable materials and treatments were combined. Twenty-seven new experimental campaigns were defined accordingly. Details about the confidence interval calculation according to LR are shown in Appendix B. Figures 15 and 16 show a comparison between the estimated slopes (and their two-sided 95% confidence intervals) with their equivalent according to gear standards for both STBF and 2T. For most cases, the standardized values lie within the confidence interval. Based on this analysis, it is not possible to state that the estimated slopes are statistically different from the ones suggested by the standards. This behavior is particularly true for \(k_{1}\) and \(k_{2}\) according to ISO 6336-3 [6] and for \(k_{1}\) and the highest (absolute) value \(k_{2}\) according to ANSI-AGMA 2001 [7]. Therefore, it can be concluded that the parameters estimated by method 2 are not different from the typical ones of case-hardened gears. As already mentioned, the FVA guideline 563 I [25] also provides indications about the allocation and numerosity of specimens. Those values have been defined within the context of data elaboration according to method 1. An indication of the number of experimental points required by model 2 can be obtained by taking inspiration from [PERSON] et al. [74]. Here, the Figure 9 database is used to evaluate the convergency of the parameter's confidence interval. Only the parameter \(s_{log(\sigma_{\text{fp}})}\) is shown here.